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Common Core: High School - Algebra : Calculate Maximum or Minimum of Quadratic by Completing the Square: CCSS.Math.Content.HSA-SSE.B.3b

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Example Question #191 : Equations / Inequalities

Complete the square to calculate the maximum or minimum point of the given function.

x^2+4x+7

Possible Answers:

$\text{Maximum}=(-2,-3)$

$\text{Maximum}=(-2,3)$

$\text{Minimum}=(-2,-3)$

$\text{Minimum}=(2,3)$

$\text{Minimum}=(-2,3)$

Correct answer:

$\text{Minimum}=(-2,3)$

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+4x+7

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+4x+7=0\Rightarrow (x+2)^2+2^2+7=0+2^2$

When simplified the new function is,

$\\(x+2)^2+4+7=4 \\(x+2)^2+11=4 \\(x+2)^2=-7$

Since the x^2 term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\\x+2=0 \\x+2-2=0-2 \\x=-2$

From here, substitute the the value into the original function.

x^2+4x+7

$\\y=(-2)^2+4(-2)+7 \\y=4-8+7 \\y=3$

Therefore the minimum value occurs at the point (-2,3) .

Report an Error

Example Question #32 : Quadratic Equations

Complete the square to calculate the maximum or minimum point of the given function.

x^2+2x-4

Possible Answers:

(-1,5)

(1,-5)

(-5,-1)

(-1,-5)

(1,5)

Correct answer:

(-1,-5)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+2x-4

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+2x-4=0\Rightarrow (x+1)^2+1^2-4=0+1^2$

When simplified the new function is,

$\\(x+1)^2-4+1 \\(x+1)^2-3$

$\\x+1=0 \\x+1-1=0-1 \\x=-1$

From here, substitute the the value into the original function.

x^2+2x-4

$\\y=(-1)^2+2(-1)-4 \\y=1-2-4 \\y=-5$

Therefore the minimum value occurs at the point (-1,-5) .

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Example Question #1 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Complete the square to calculate the maximum or minimum point of the given function.

x^2+8x-3

Possible Answers:

(4,-19)

(-19,-4)

(-4,19)

(-4,-19)

(4,19)

Correct answer:

(-4,-19)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+8x-3

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$x^2+8x-3\Rightarrow (x+4)^2-3+4^2$

When simplified the new function is,

$\\(x+4)^2-3+16\\(x+4)^2+13$

$\\x+4=0 \\x+4-4=0-4 \\x=-4$

From here, substitute the the value into the original function.

x^2+8x-3

$\\y=(-4)^2+8(-4)-3 \\y=16-32-3 \\y=-19$

Therefore the minimum value occurs at the point (-4,-19) .

Report an Error

Example Question #51 : Seeing Structure In Expressions

Complete the square to calculate the maximum or minimum point of the given function.

x^2+6x+2

Possible Answers:

(-3,7)

(-7,-3)

(3,7)

(3,-7)

(-3,-7)

Correct answer:

(-3,-7)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+6x+2

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$x^2+6x+2\Rightarrow (x+3)^2+6+3^2$

When simplified the new function is,

$\\(x+3)^2+2+9\\(x+3)^2+11$

$\\x+3=0 \\x+3-3=0-3 \\x=-3$

From here, substitute the the value into the original function.

x^2+6x+2

$\\y=(-3)^2+6(-3)+2 \\y=9-18+2 \\y=-7$

Therefore the minimum value occurs at the point (-3,-7) .

Report an Error

Example Question #2 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Complete the square to calculate the maximum or minimum point of the given function.

x^2-4x+1

Possible Answers:

(-2,3)

(-2,-3)

(2,3)

(2,-3)

(-3,2)

Correct answer:

(2,-3)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2-4x+1

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -4$

Now divide the middle term coefficient by two.

$\frac{-4}{2}=-2$

From here write the function with the perfect square.

$x^2-4x+1\Rightarrow (x-2)^2+1+(-2)^2$

When simplified the new function is,

$\\(x-2)^2+1+4\\(x-2)^2+5$

$\\x-2=0 \\x-2+2=0+2\\x=2$

From here, substitute the the value into the original function.

x^2-4x+1

$\\y=(2)^2-4(2)+1 \\y=4-8+1 \\y=-3$

Therefore the minimum value occurs at the point (2,-3) .

Report an Error

Example Question #6 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Complete the square to calculate the maximum or minimum point of the given function.

x^2-6x-6

Possible Answers:

(-3,-15)

(-3,15)

(-15,3)

(3,15)

(3,-15)

Correct answer:

(3,-15)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2-6x-6

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -6$

Now divide the middle term coefficient by two.

$\frac{-6}{2}=-3$

From here write the function with the perfect square.

$x^2-6x-6\Rightarrow (x-3)^2-6+(-3)^2$

When simplified the new function is,

$\\(x-3)^2-6+9\\(x-3)^2+3$

$\\x-3=0 \\x-3+3=0+3\\x=3$

From here, substitute the the value into the original function.

x^2-6x-6

$\\y=(3)^2-6(3)-6 \\y=9-18-6 \\y=-15$

Therefore the minimum value occurs at the point (3,-15) .

Report an Error

Example Question #51 : Seeing Structure In Expressions

Complete the square to calculate the maximum or minimum point of the given function.

-x^2-2x-1

Possible Answers:

(-1,0)

(0,0)

(0,-1)

(1,0)

(0,1)

Correct answer:

(-1,0)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

-x^2-2x-1

First factor out a negative one.

-1(x^2+2x+1)

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square.

$-(x^2+2x+1)\Rightarrow-((x+1)^2+1+1)$

When simplified the new function is,

$\\-((x+1)^2+2)$

Since the x^2 term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\\x+1=0 \\x+1-1=0-1 \\x=-1$

From here, substitute the the value into the original function.

-x^2-2x-1

$\\y=-(-1)^2-2(-1)-1 \\y=-1+2-1 \\y=0$

Therefore the maximum value occurs at the point (-1,0) .

Report an Error

Example Question #1 : Graphing Quadratics & Polynomials

Complete the square to calculate the maximum or minimum point of the given function.

-x^2-4x-1

Possible Answers:

(-2,-3)

(2,3)

(2,-3)

(3,-2)

(-2,3)

Correct answer:

(-2,3)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

-x^2-4x-1

First factor out a negative one.

-1(x^2+4x+1)

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+1)\Rightarrow-((x+2)^2+1+2^2)$

When simplified the new function is,

$\\-((x+2)^2+5)$

$\\x+2=0 \\x+2-2=0-2 \\x=-2$

From here, substitute the the value into the original function.

-x^2-4x-1

$\\y=-(-2)^2-4(-2)-1 \\y=-4+8-1 \\y=3$

Therefore the maximum value occurs at the point (-2,3) .

Report an Error

Example Question #551 : New Sat

Complete the square to calculate the maximum or minimum point of the given function.

-x^2-6x-1

Possible Answers:

(-3,-8)

(3,-8)

(8,-3)

(3,8)

(-3,8)

Correct answer:

(-3,8)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

-x^2-6x-1

First factor out a negative one.

-1(x^2+6x+1)

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$-(x^2+6x+1)\Rightarrow-((x+3)^2+1+3^2)$

When simplified the new function is,

$\\-((x+3)^2+10)$

$\\x+3=0 \\x+3-3=0-3 \\x=-3$

From here, substitute the the value into the original function.

-x^2-6x-1

$\\y=-(-3)^2-6(-3)-1 \\y=-9+18-1 \\y=8$

Therefore the maximum value occurs at the point (-3,8) .

Report an Error

Example Question #52 : Seeing Structure In Expressions

Complete the square to calculate the maximum or minimum point of the given function.

-x^2-4x-5

Possible Answers:

(2,1)

(2,-1)

(-2,-1)

(-2,1)

(-1,-2)

Correct answer:

(-2,-1)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

-x^2-4x-5

First factor out a negative one.

-1(x^2+4x+5)

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+5)\Rightarrow-((x+2)^2+5+2^2)$

When simplified the new function is,

$\\-((x+2)^2+9)$

$\\x+2=0 \\x+2-2=0-2 \\x=-2$

From here, substitute the the value into the original function.

-x^2-4x-5

$\\y=-(-2)^2-4(-2)-5 \\y=-4+8-5 \\y=-1$

Therefore the maximum value occurs at the point (-2,-1) .

Report an Error

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Common Core: High School - Algebra : Calculate Maximum or Minimum of Quadratic by Completing the Square: CCSS.Math.Content.HSA-SSE.B.3b

Study concepts, example questions & explanations for Common Core: High School - Algebra

All Common Core: High School - Algebra Resources

Example Questions

Example Question #191 : Equations / Inequalities

Example Question #32 : Quadratic Equations

Example Question #1 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #51 : Seeing Structure In Expressions

Example Question #2 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #6 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #51 : Seeing Structure In Expressions

Example Question #1 : Graphing Quadratics & Polynomials

Example Question #551 : New Sat

Example Question #52 : Seeing Structure In Expressions

All Common Core: High School - Algebra Resources

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