9.8 Solve Quadratic Inequalities

Before you get started, take this readiness quiz.

Solve: $2 x - 3 = 0 .$
If you missed this problem, review the reference.

Solve: $2 y^{2} + y = 15$.
If you missed this problem, review the reference.

Solve $\frac{1}{x^{2} + 2 x - 8} > 0$
If you missed this problem, review the reference.

Solve $x^{2} + 2 x - 8 \geq 0$ algebraically. Write the solution in interval notation.

Solve $x^{2} - 2 x - 15 \leq 0$ algebraically. Write the solution in interval notation.

Solve $− x^{2} + 2 x + 1 \geq 0$ algebraically. Write the solution in interval notation.

Solve $− x^{2} + 8 x - 14 < 0$ algebraically. Write the solution in interval notation.

Solve and write any solution in interval notation:
ⓐ $− x^{2} + 2 x - 4 \leq 0$ ⓑ $− x^{2} + 2 x - 4 \geq 0$

Solve and write any solution in interval notation:
ⓐ $x^{2} + 3 x + 3 < 0$ ⓑ $x^{2} + 3 x + 3 > 0$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} + 3 x - 4 \geq 0$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 7 x + 10 < 0$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} + 8 x > - 15$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 4 x + 2 \leq 0$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 10 x > - 19$

Solve each inequality algebraically and write any solution in interval notation.

$−6 x^{2} + 19 x - 10 \geq 0$

Solve each inequality algebraically and write any solution in interval notation.

$−2 x^{2} + 7 x + 4 \geq 0$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} + 3 x + 5 > 0$

Solve each inequality algebraically and write any solution in interval notation.

$− x^{2} + x - 7 > 0$

Solve each inequality algebraically and write any solution in interval notation.

$−2 x^{2} + 8 x - 10 < 0$

Explain zero partition numbers and how they are used to solve quadratic inequalities algebraically.

Describe the steps needed to solve a quadratic inequality graphically.

Solve using the Square Root Property.

$y^{2} = 144$

Solve using the Square Root Property.

$4 a^{2} = 100$

Solve using the Square Root Property.

$r^{2} + 32 = 0$

Solve using the Square Root Property.

$\frac{2}{3} w^{2} - 20 = 30$

Solve using the Square Root Property.

$\left(p - 5\right)^{2} + 3 = 19$

Solve using the Square Root Property.

$\left(x - \frac{1}{4}\right)^{2} = \frac{3}{16}$

Solve using the Square Root Property.

$\left(n - 4\right)^{2} - 50 = 150$

Solve using the Square Root Property.

$n^{2} + 10 n + 25 = 12$

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$x^{2} + 22 x$

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$a^{2} - 3 a$

Solve by completing the square.

$d^{2} + 14 d = −13$

Solve by completing the square.

$m^{2} + 6 m = −109$

Solve by completing the square.

$v^{2} - 14 v = −31$

Solve by completing the square.

$m^{2} + 10 m - 4 = −13$

Solve by completing the square.

$a^{2} = 3 a + 8$

Solve by completing the square.

$\left(u + 8\right) \left(u + 4\right) = 14$

Solve by completing the square.

$3 p^{2} - 18 p + 15 = 15$

Solve by completing the square.

$4 y^{2} - 6 y = 4$

Solve by completing the square.

$3 c^{2} + 2 c = 9$

Solve by completing the square.

$2 x^{2} + 6 x = −5$

Solve by using the Quadratic Formula.

$4 x^{2} - 5 x + 1 = 0$

Solve by using the Quadratic Formula.

$r^{2} - r - 42 = 0$

Solve by using the Quadratic Formula.

$4 v^{2} + v - 5 = 0$

Solve by using the Quadratic Formula.

$3 m^{2} + 8 m + 2 = 0$

Solve by using the Quadratic Formula.

$6 a^{2} - 5 a + 2 = 0$

Solve by using the Quadratic Formula.

$u \left(u - 10\right) + 3 = 0$

Solve by using the Quadratic Formula.

$\frac{1}{8} p^{2} - \frac{1}{5} p = - \frac{1}{20}$

Solve by using the Quadratic Formula.

$4 c^{2} + 4 c + 1 = 0$

Determine the number of solutions for each quadratic equation.

ⓐ $9 x^{2} - 6 x + 1 = 0$
ⓑ $3 y^{2} - 8 y + 1 = 0$
ⓒ $7 m^{2} + 12 m + 4 = 0$
ⓓ $5 n^{2} - n + 1 = 0$

Identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

ⓐ $16 r^{2} - 8 r + 1 = 0$
ⓑ $5 t^{2} - 8 t + 3 = 9$
ⓒ $3 \left(c + 2\right)^{2} = 15$

Solve.

$x^{4} - 14 x^{2} + 24 = 0$

Solve.

$4 x^{4} - 5 x^{2} + 1 = 0$

Solve.

$x + 3 \sqrt{x} - 28 = 0$

Solve.

$x^{\frac{2}{3}} - 10 x^{\frac{1}{3}} + 24 = 0$

Solve.

$8 x^{−2} - 2 x^{−1} - 3 = 0$

Solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Find two consecutive even numbers whose product is 624.

Solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.

Solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

Solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the w…

Solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

Determine if the following parabolas open up or down.

ⓐ $y = x^{2} + 8 x - 1$
ⓑ $y = −4 x^{2} - 7 x + 1$

Find ⓐ the equation of the axis of symmetry and ⓑ the vertex.

$y = 2 x^{2} - 8 x + 1$

Find the x- and y-intercepts.

$y = x^{2} - 8 x + 15$

Find the x- and y-intercepts.

$y = −5 x^{2} - 30 x - 46$

Find the x- and y-intercepts.

$y = x^{2} + 16 x + 64$

Find the minimum or maximum value.

$y = −3 x^{2} + 12 x - 10$

Rewrite each function in the $f \left(x\right) = a \left(x - h\right)^{2} + k$ form by completing the square.

$f \left(x\right) = 2 x^{2} - 4 x - 4$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 6 x + 8 < 0$

Solve each inequality algebraically and write any solution in interval notation.

$x^{2} - 6 x + 4 \leq 0$

Solve each inequality algebraically and write any solution in interval notation.

$− x^{2} + x - 6 > 0$

Use the Square Root Property to solve the quadratic equation $3 \left(w + 5\right)^{2} = 27 .$

Use the Quadratic Formula to solve the quadratic equation $2 m^{2} - 5 m + 3 = 0 .$

Solve the following quadratic equations. Use any method.

$\frac{9}{4} y^{2} - 3 y + 1 = 0$

Use the discriminant to determine the number and type of solutions of each quadratic equation.

$3 q^{2} - 10 q + 12 = 0$

Solve each equation.

$y^{\frac{2}{3}} + 2 y^{\frac{1}{3}} - 3 = 0$

For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.

$y = - x^{2} - 8 x - 16$

Solve each inequality algebraically and write any solution in interval notation.

$2 x^{2} + x - 10 > 0$

Model the situation with a quadratic equation and solve by any method.

The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)