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Algebra II : Quadratic Equations and Inequalities

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Example Questions

Example Question #45 : Quadratic Formula

Solve for the roots: y=8x^2-4x-2

Possible Answers:

$\frac{1\pm\sqrt{5}}{4}$

$\frac{-1\pm\sqrt{5}}{2}$

$\frac{1\pm\sqrt{5}}{16}$

$\frac{-1\pm i\sqrt{5}}{4}$

$\frac{-1\pm\sqrt{5}}{4}$

Correct answer:

$\frac{1\pm\sqrt{5}}{4}$

Explanation:

Write the quadratic formula.

$x= \frac{-b\pm\sqrt{b^2-4ac}}{2a}$

The given equation y=8x^2-4x-2 is in standard form of:

a=ax^2+bx+c

The coefficients correspond to the values that go inside the quadratic equation.

a=8, b=-4, c=-2

Substitute the values into the equation.

$x= \frac{-(-4)\pm\sqrt{(-4)^2-4(8)(-2)}}{2(8)}$

Simplify this equation.

$x= \frac{4\pm\sqrt{16+64}}{16} = \frac{4\pm\sqrt{80}}{16}$

The radical can be rewritten as:

$\sqrt{80} = \sqrt{16}\cdot \sqrt 5 = 4\sqrt5$

Substitute and simplify the fraction.

$x=\frac{4\pm4\sqrt{5}}{16}=\frac{1\pm\sqrt{5}}{4}$

The answer is: $\frac{1\pm\sqrt{5}}{4}$

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Example Question #46 : Quadratic Formula

Solve for the roots, if any: y= -8x^2-x-1

Possible Answers:

$-\frac{1}{4}\pm\frac{i\sqrt{31}}{4}$

$-\frac{1}{16}\pm\frac{i\sqrt{33}}{16}$

$\textup{No solution.}$

$-\frac{1}{8}\pm\frac{i\sqrt{33}}{8}$

$-\frac{1}{16}\pm\frac{i\sqrt{31}}{16}$

Correct answer:

$-\frac{1}{16}\pm\frac{i\sqrt{31}}{16}$

Explanation:

This polynomial is in the standard form of a parabola, y=ax^2+bx+c .

Write the quadratic formula.

$x= \frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Identify the variables.

a=-8, b=-1, c=-1

$x= \frac{-(-1)\pm\sqrt{(-1)^2-4(-8)(-1)}}{2(-8)}$

Simplify the quadratic.

$x= \frac{1\pm\sqrt{1-32}}{-16}=\frac{1\pm\sqrt{-31}}{-16}= \frac{1\pm i\sqrt{31}}{-16}$

Split the fraction.

The answer is: $-\frac{1}{16}\pm\frac{i\sqrt{31}}{16}$

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Example Question #501 : Intermediate Single Variable Algebra

Determine the roots, if possible: y=15x^2-x-1

Possible Answers:

$\frac{1\pm i\sqrt{31}}{15}$

$\textup{There are no possible roots.}$

$\frac{1\pm\sqrt{61}}{30}$

$\frac{1\pm i\sqrt{61}}{30}$

$\pm\frac{2 i\sqrt{15}}{15}$

Correct answer:

$\frac{1\pm\sqrt{61}}{30}$

Explanation:

Write the quadratic equation.

$x= \frac{-b\pm\sqrt{b^2-4ac}}{2a}$

This equation will apply for polynomials in y=ax^2+bx+c format.

$y=15x^2-x-1 \rightarrow a=15, b=-1, c=-1$

Substitute the correct values into the quadratic formula.

$x= \frac{-(-1)\pm\sqrt{(-1)^2-4(15)(-1)}}{2(15)}$

Simplify this equation.

$x= \frac{1\pm\sqrt{1+60}}{30} = \frac{1\pm\sqrt{61}}{30}$

The answer is: $\frac{1\pm\sqrt{61}}{30}$

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Example Question #48 : Quadratic Formula

A ball is launched straight up from the ground at time t=0 , and the height of the ball from the ground at time is described by the function h(t)=-3x^2 +62x . Find the time at which the ball returns to the ground.

Possible Answers:

t=16

$t=\frac{67}{3}$

$t=\frac{62}{3}$

$t=\frac{60}{3}$

t=15

Correct answer:

$t=\frac{62}{3}$

Explanation:

In order to solve this problem, we think about how to translate what is asked of us in the problem into a mathematical notion. The problem asks us to find when the ball returns to the ground. We know that the ball is launched from the ground at time t=0 , and if we plug t=0 into our original function, we will find that at time t=0 , h(0)=(0)^2-8(0)=0 . Therefore, our "ground" is located at h(t)=0 , which is also the horizontal axis. This means that if we are looking for when our ball returns to the ground, mathematically we are looking for values of which make h(t)=0 . In other words, we are looking for the zeros (or horizontal intercepts) of our function.

Typically, the easiest way to find zeros is to factor, but in this situation factoring won't get us very far. So, we have to turn the quadratic formula. Remember for a function f(x) in the standard quadratic form, f(x)=ax^2+bx+c , you can find the zeros by plugging them into the quadratic formula, which is...

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

The first step in using the quadratic formula is to identify the proper , , and . In our case, a=-3 and b=62 , while c=0 . So, to solve for our zeros, we simply need to plug these numbers into the quadratic formula and solve for x.

$t=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$t=\frac{-(62)\pm \sqrt{(-62)^2-4(-3)(0)}}{2(-3)}$

$t=\frac{-62\pm \sqrt{3844}}{-6}$

$t=\frac{-62\pm 62}{-6}$

$t=\frac{0}{-6},\frac{-124}{-6}$

$t=0,\frac{62}{3}$

Since we know that t=0 is when our ball was launched, we know that $t=\frac{62}{3}$ must be when our ball returns to the ground.

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Example Question #502 : Intermediate Single Variable Algebra

Find the zeros of the function f(x) where

f(x)=6x^2+11x-35

Possible Answers:

$x=-\frac{7}{2},\frac{5}{3}$

$x=\frac{7}{3},\frac{5}{4}$

$x=\frac{7}{2},\frac{5}{3}$

$x=-\frac{7}{3},-\frac{5}{4}$

$x=\frac{1}{3},-\frac{1}{2}$

Correct answer:

$x=-\frac{7}{2},\frac{5}{3}$

Explanation:

Typically, the easiest way to find the zeros of a function is to factor. However, with this quadratic, which may not be particularly easy to factor by hand, quadratic formula may be easier. Recall that for a polynomial in the standard quadratic form, f(x)=ax^2+bx+c , the quadratic formula will give you all real zeros of the polynomial. First, we need to identify what , , and are in our original function. In this case, a=6 , b=11 , and c=-35 . Next, we need to set up the quadratic formula, plug in our values, and solve.

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$x=\frac{-11\pm \sqrt{(11)^2-4(6)(-35)}}{2(6)}$

$x=\frac{-11\pm \sqrt{121+840}}{12}$

$x=\frac{-11\pm \sqrt{961}}{12}$

$x=\frac{-11\pm 31}{12}$

$x=-\frac{42}{12},\frac{20}{12}$

$x=-\frac{7}{2},\frac{5}{3}$

Now we have our zeros!

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Example Question #50 : Quadratic Formula

Determine a root: y=-3x^2-x-1

Possible Answers:

$-\frac{1}{6}+\frac{i\sqrt{22}}{6}$

$-\frac{1}{6}+\frac{2i\sqrt{6}}{3}$

$-\frac{1}{3}+\frac{i\sqrt{11}}{3}$

$-\frac{1}{6}+\frac{i\sqrt{11}}{6}$

$-\frac{1}{2}+\frac{i\sqrt{11}}{2}$

Correct answer:

$-\frac{1}{6}+\frac{i\sqrt{11}}{6}$

Explanation:

Identify the coefficients given the equation in standard form:

y=ax^2+bx+c

a=-3, b=-1,c=-1

Write the quadratic formula.

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substitute the coefficients.

$x=\frac{-(-1)\pm \sqrt{(-1)^2-4(-3)(-1)}}{2(-3)}$

$x=\frac{1\pm \sqrt{1-12}}{-6} = \frac{1\pm \sqrt{-11}}{-6} = \frac{1\pm i\sqrt{11}}{-6}$

These two roots are complex and either is a valid answer.

The answer is: $-\frac{1}{6}+\frac{i\sqrt{11}}{6}$

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Example Question #191 : Solving Quadratic Equations

Determine a possible root for: y=x^2+10x-1

Possible Answers:

$5+3\sqrt{2}$

$\sqrt{13}-5$

$\sqrt{26}-5$

$\textup{None of the answers are possible roots.}$

$5-2\sqrt6$

Correct answer:

$\sqrt{26}-5$

Explanation:

Write the quadratic formula.

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Identify and substitute the terms.

a=1,b=10, c=-1

$x=\frac{-10\pm \sqrt{10^2-4(1)(-1)}}{2(1)}=\frac{-10\pm \sqrt{100+4}}{2}$

$x=\frac{-10\pm \sqrt{104}}{2}$

Factor the radical using factors of perfect squares.

$\sqrt{104} = \sqrt4 \cdot \sqrt{26} = 2\sqrt{26}$

$x=\frac{-10\pm2 \sqrt{26}}{2} = -5\pm \sqrt{26}$

These two answers are possible roots.

The answer is: $\sqrt{26}-5$

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Example Question #511 : Intermediate Single Variable Algebra

Find the roots of the quadratic function,

f(x)= -A^2x^2+2Ax+1

Where is any real number constant not equal to zero.

Possible Answers:

$x = \left \{\frac{1}{3}-\frac{\sqrt{3}}{A}i\: \: \: ,\: \: \: \frac{1}{3}+\frac{\sqrt{3}}{A}i\right \}$

$x = \left \{\frac{1}{A^2}-\frac{3\sqrt{3}}{A^2}\: \: \: ,\: \: \: \frac{1}{A^2}+\frac{3\sqrt{3}}{A^2}\right \}$

$x = \left \{\frac{1}{A^2}-\frac{2\sqrt{2}}{A}\: \: \: ,\: \: \: \frac{1}{A^2}+\frac{2\sqrt{2}}{A}\right \}$

$x = \left \{\frac{1}{A}-\frac{\sqrt{2}}{A}\: \: \: ,\: \: \: \frac{1}{A}+\frac{\sqrt{2}}{A}\right \}$

$x = \left \{\frac{1}{A}-\frac{\sqrt{2}}{A}i\: \: \: ,\: \: \: \frac{1}{A}+\frac{\sqrt{2}}{A}i\right \}$

Correct answer:

$x = \left \{\frac{1}{A}-\frac{\sqrt{2}}{A}\: \: \: ,\: \: \: \frac{1}{A}+\frac{\sqrt{2}}{A}\right \}$

Explanation:

f(x)= -A^2x^2+2Ax+1

To find the roots set the function to zero:

f(x)=0 ,

-A^2x^2 +2Ax+1=0 (1)

Apply the quadratic formula:

_________________________________________________________________

Reminder

Recall that for a quadratic ax^2 +bx+c=0 the general formula for the solution in terms of the constant coefficients is given by:

$x = \frac{-b\pm \sqrt{b^2-4ca}}{2a}$ (2)

_____________________________________________________________

Use equation (2) to write a solution for equation (1).

$x = \frac{-2A \pm \sqrt{4A^2-4(-A^2)}}{2(-A^2)}$

$x = \frac{2A \pm \sqrt{8A^2}}{2A^2}$

If we simplify the right-hand term in the numerator we obtain:

$\sqrt{8A^2}=2\sqrt2A$

So now we have for ,

$x=\frac{2A}{2A^2} \pm \frac{2\sqrt{2}A}{2A^2}$

After all the cancellations in the expression above we obtain:

$x = \frac{1}{A} \pm \frac{\sqrt{2}}{A}$

Therefore, the solution set for this equation is:

$x = \left \{\frac{1}{A}-\frac{\sqrt{2}}{A}\: \: \: ,\: \: \: \frac{1}{A}+\frac{\sqrt{2}}{A}\right \}$

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Example Question #193 : Solving Quadratic Equations

Find the roots of the quadratic function,

f(x)=3x^2+7x+14

Possible Answers:

$x = \left \{ \frac{-7+\sqrt{117}i}{6}, \frac{-7-\sqrt{117}i}{6}\right \},$

$x = \left \{ \frac{-7+\sqrt{119}i}{6}, \frac{-7-\sqrt{119}i}{6}\right \},$

$x = \left \{ \frac{7+\sqrt{119}}{6}, \frac{7-\sqrt{119}}{6}\right \},$

$x = \left \{ \frac{-3+\sqrt{217}i}{2}, \frac{-3-\sqrt{217}i}{2}\right \},$

$x = \left \{ \frac{-3+\sqrt{217}}{2}, \frac{-3-\sqrt{217}}{2}\right \},$

Correct answer:

$x = \left \{ \frac{-7+\sqrt{119}i}{6}, \frac{-7-\sqrt{119}i}{6}\right \},$

Explanation:

f(x)=3x^2+7x+14

The roots are the values of for which:

f(x)=0

3x^2+7x+14=0

______________________________________________________________

Reminder

Recall that for a quadratic ax^2 +bx+c=0 the general formula for the solution in terms of the constant coefficients is given by:

$x = \frac{-b\pm \sqrt{b^2-4ca}}{2a}$

_____________________________________________________________

Use the quadratic formula to find the roots.

$x = \frac{-7\pm\sqrt{7^2-4(14)(3)})}{2(3)}$

$x = \frac{7}{6}\pm \frac{\sqrt{-119}}{6}$

Notice that $\sqrt{-119}$ is not a real number, and therefore the roots will be complex numbers.

Using the definition of the imaginary unit $i = \sqrt{-1}$ we can rewrite $\sqrt{-119}$ as follows,

$\sqrt{-119}=\sqrt{119(-1)}=\sqrt{119}\sqrt{-1}=\sqrt{119}i$

Now we can write the solutions to this problem in the form:

$x = \left \{ \frac{7+\sqrt{119}}{6}i, \frac{7-\sqrt{119}}{6}i\right \},$

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Example Question #341 : Quadratic Equations And Inequalities

Find the roots using the quadratic formula.

x^2+9x+15

Possible Answers:

x= 2.21

x= -6.79

x= 6.79, 2.21

x= -6.79,-2.21

x= 7.89,5.65

Correct answer:

x= -6.79,-2.21

Explanation:

$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

For this problem

a=1, coefficient of x^2 term

b=9, the coefficient of the x term

c=15, the constant term

$\frac{-9\pm \sqrt{9^2-4(1)(15)}}{2(1)}=\frac{-9\pm \sqrt{21}}{2}=\frac{-9}{2}\pm \frac{\sqrt{21}}{2}$

solving the expression shows the roots of -6.79 and -2.21

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Algebra II : Quadratic Equations and Inequalities

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #45 : Quadratic Formula

Example Question #46 : Quadratic Formula

Example Question #501 : Intermediate Single Variable Algebra

Example Question #48 : Quadratic Formula

Example Question #502 : Intermediate Single Variable Algebra

Example Question #50 : Quadratic Formula

Example Question #191 : Solving Quadratic Equations

Example Question #511 : Intermediate Single Variable Algebra

Example Question #193 : Solving Quadratic Equations

Example Question #341 : Quadratic Equations And Inequalities

All Algebra II Resources

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