2.5 Quadratic Equations

One rational solution (double solution)

b2 − 4ac = 0

Two irrational solutions

b2 − 4ac > 0, not a perfect square

Two rational solutions

b2 − 4ac > 0, perfect square

Using the Quadratic Formula

1. First, move the constant term to the right side of the equal sign: ax 2 + bx = −c 2. As we want the leading coeffici t to equal 1, divide through by a: x 2 + _ab x = − a_c 3. Then, fi d 1 2 of the middle term, and add 1 __ 2 _b_ a 2 = b2 _ 4a2 to both sides of the equal sign: x2 + _ab x + b2 _ 4a2 = b2 _ 4a2 − a_c 4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction: x + _b 2a 2 = b_2 − 4ac 4a2 5. Now, use the square root property, which gives x + _b 2a = °æ √ ________ b 2 − 4ac _ 4a2 x + _b 2a = °æ √— _b2 −_ 4ac 2a 6. Finally, add − _b 2a to both sides of the equation and combine the terms on the right side. Thus, x = −b °æ √— _b2 − 4ac 2a

the discriminant

For ax2 + bx + c = 0, where a, b, and c are rational and real numbers, the discriminant is the expression under the radical in the quadratic formula: b2 − 4ac. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Solving Quadratics with a Leading Coefficient of 1

Given a quadratic equation with the leading coefficient of 1, factor it. 1. Find two numbers whose product equals c and whose sum equals b. 2. Use those numbers to write two factors of the form (x + k) or (x − k), where k is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and −2, the factors are (x + 1)(x − 2). 3. Solve using the zero-product property by setting each factor equal to zero and solving for the variable.

the zero-product property and quadratic equations

The zero-product property states If a ⋅ b = 0, then a = 0 or b = 0, where a and b are real numbers or algebraic expressions. A quadratic equation is an equation containing a second-degree polynomial; for example ax 2 + bx + c = 0 where a, b, and c are real numbers, and if a ≠ 0, it is in standard form.

Completing the Square

Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square. We will use the example x2 + 4x + 1 = 0 to illustrate each step. 1. Given a quadratic equation that cannot be factored, and with a = 1, fi st add or subtract the constant term to the right sign of the equal sign. x 2 + 4x = −1 2. Multiply the b term by 1/2 and square it. 1 2 (4) = 2 22 = 4 3. Add 1/2 b ^ 2 to both sides of the equal sign and simplify the right side. We have x 2 + 4x + 4 = −1 + 4 x 2 + 4x + 4 = 3 4. The left side f the equation can now be factored as a perfect square. x 2 + 4x + 4 = 3 (x + 2)2 = 3 5. Use the square root property and solve. √—(x + 2)2 = °æ √ —3 x + 2 = °æ √—3 x = −2 °æ √—3 6. The solutions are −2 + √—3 a nd −2 − √—3 .

Factoring and Solving a Quadratic Equation of Higher Order

When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let's review the grouping procedures: 1. With the quadratic in standard form, ax2 + bx + c = 0, multiply a ⋅ c. 2. Find two numbers whose product equals ac and whose sum equals b. 3. Rewrite the equation replacing the bx term with two terms using the numbers found in step 1 as coeffici ts of x. 4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping. 5. Factor out the expression in parentheses. 6. Set the expressions equal to zero and solve for the variable.

imaginary solutions

When you get i as part of your solution (no real part)

Two complex solutions

b2 − 4ac < 0

Quadratic equation

ax² + bx + c = 0

the square root property

if x 2 = k, then x = +/- √k

the quadratic formula

x = -b ± √(b² - 4ac)/2a