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PSAT Math : Quadratic Equations

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Example Questions

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Example Question #261 : Algebra

36x² -12x - 15 = 0

Solve for x

Possible Answers:

-1/2 and -5/6

1/2 and 1/3

1/2 and -1/3

-1/2 and 5/6

1/2 and 5/6

Correct answer:

-1/2 and 5/6

Explanation:

36x² - 12x - 15 = 0

Factor the equation:

(6x + 3)(6x - 5) = 0

Set each side equal to zero

6x + 3 = 0

x = -3/6 = -1/2

6x – 5 = 0

x = 5/6

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Example Question #31 : Quadratic Equations

$\textup{If }x-y=5\textup{ and }x^{2}-y^{2}=35,\textup{ then }xy=$

Possible Answers:

Correct answer:

Explanation:

$x^{2}-y^{2}=(x-y)(x+y)\textup{ Substitute. Given: }x-y = 5,\textup{ so:}$

$5(x+y)=35\:,\:x+y = 7 ,\: \textup{Combine equations: } x+y+x-y=7+5$

$2x = 12\:,\:x=6,\:y=1,\: \:(6)(1)=6$

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Example Question #31 : Quadratic Equations

Factor the following:

x^2+11x+30

Possible Answers:

(x + 5)(x+6)

(x + 15)(x+2)

(x - 5)(x-6)

(x +30)(x-1)

(x + 3)(x+10)

Correct answer:

(x + 5)(x+6)

Explanation:

Since you have an addition for the last element, you know that you will have either two subtractions or two additions in your factored groups. Since the second term is positive, your factored groups will contain two additions:

(x+?)(x+?)

Now, all you have to do are go through the factors of 30 and find the pair that adds up to 11:

and : No!

and : No! (But closer!)

and : Yes!

Therefore, your answer will be:

(x + 5)(x+6)

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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) .

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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)

(4,-19)

(-19,-4)

(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) .

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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) .

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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) .

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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) .

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PSAT Math : Quadratic Equations

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #261 : Algebra

Example Question #31 : Quadratic Equations

Example Question #31 : Quadratic Equations

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

All PSAT Math Resources

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