2.5 Quadratic Equations

Recognize and use the appropriate method to factor a polynomial completely.

1. ⓐ $5 x^{3} - 15 x^{2} - 50 x$
2. ⓑ $4 x^{2} + 20 x y + 25 y^{2}$
3. ⓒ $9 x^{2} - 16$
4. ⓓ $8 x^{3} + 27 y^{3}$
5. ⓔ $16 x^{2} + 24 x y - 4 x - 6 y$

Factor and solve the quadratic equation: $x^{2} - 5 x - 6 = 0.$

Solve the quadratic equation by factoring: $x^{2} + 8 x + 15 = 0.$

Solve the quadratic equation by factoring: $x^{2} - 4 x - 21 = 0.$

Solve the difference of squares equation using the zero-product property: $x^{2} - 9 = 0.$

Solve by factoring: $x^{2} - 25 = 0.$

Solve using factoring by grouping: $12 x^{2} + 11 x + 2 = 0.$

Solve the equation by factoring: $−3 x^{3} - 5 x^{2} - 2 x = 0.$

Solve by factoring: $x^{3} + 11 x^{2} + 10 x = 0.$

Solve the quadratic using the square root property: $x^{2} = 8.$

Solve the quadratic equation: $4 x^{2} + 1 = \text{7}.$

Solve the quadratic equation using the square root property: $3 \left(x - 4\right)^{2} = 15.$

Solve the quadratic equation by completing the square: $x^{2} - 3 x - 5 = 0.$

Solve by completing the square: $x^{2} - 6 x = 13.$

Solve the quadratic equation: $x^{2} + 5 x + 1 = 0.$

Use the quadratic formula to solve $x^{2} + x + 2 = 0.$

Solve the quadratic equation using the quadratic formula: $9 x^{2} + 3 x - 2 = 0.$

Use the discriminant to find the nature of the solutions to the following quadratic equations:

1. ⓐ $x^{2} + 4 x + 4 = 0$
2. ⓑ $8 x^{2} + 14 x + 3 = 0$
3. ⓒ $3 x^{2} - 5 x - 2 = 0$
4. ⓓ $3 x^{2} - 10 x + 15 = 0$

Use the Pythagorean Theorem to solve the right triangle problem: Leg a measures 4 units, leg b measures 3 units. Find the length of the hypotenuse.

How do we recognize when an equation is quadratic?

When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?

Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.

Solve the quadratic equation by factoring.

$x^{2} - 9 x + 18 = 0$

Solve the quadratic equation by factoring.

$6 x^{2} + 17 x + 5 = 0$

Solve the quadratic equation by factoring.

$3 x^{2} - 75 = 0$

Solve the quadratic equation by factoring.

$4 x^{2} = 9$

Solve the quadratic equation by factoring.

$5 x^{2} = 5 x + 30$

Solve the quadratic equation by factoring.

$7 x^{2} + 3 x = 0$

Solve the quadratic equation by using the square root property.

$x^{2} = 36$

Solve the quadratic equation by using the square root property.

$\left(x - 1\right)^{2} = 25$

Solve the quadratic equation by using the square root property.

$\left(2 x + 1\right)^{2} = 9$

Solve the quadratic equation by completing the square. Show each step.

$x^{2} - 9 x - 22 = 0$

Solve the quadratic equation by completing the square. Show each step.

$x^{2} - 6 x = 13$

Solve the quadratic equation by completing the square. Show each step.

$2 + z = 6 z^{2}$

Solve the quadratic equation by completing the square. Show each step.

$2 x^{2} - 3 x - 1 = 0$

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$x^{2} + 4 x + 7 = 0$

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$9 x^{2} - 30 x + 25 = 0$

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.

$6 x^{2} - x - 2 = 0$

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$x^{2} + x = 4$

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

$3 x^{2} - 5 x + 1 = 0$

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

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

Enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thous…

Enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2^nd CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thous…

Beginning with the general form of a quadratic equation, $a x^{2} + b x + c = 0 ,$ solve for x by using the completing the square method, thus deriving the quadratic formula.

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft.². Solve the quadratic equation to find the length and width.

Suppose that an equation is given $p = −2 x^{2} + 280 x - 1000 ,$ where $x$ represents the number of items sold at an auction and $p$ is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2^nd CALC maximum. To obtain a good window for the curve, set $x$ [0,200] and $y$ [0,10000].

The cost function for a certain company is $C = 60 x + 300$ and the revenue is given by $R = 100 x - 0.5 x^{2} .$ Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of $300.