Algebra 1 Solving Quadratic Equations

Quadratic Form

y = ax² + bx + c

Completing the Square

(1/2*b)² Perform prior to solving the equation (1/2*10)² Substitute in (5)² Solve 25 Example: x² + 10x - 11 = 0 + 11 + 11 Add c to both sides _______________________ x² + 10x = 11 Add previous answer to both sides + 25 + 25 _______________________ x² + 10x + 25 = 36 Factor the equation 1* 25 & 5*5 (x + 5)(x +5) = 36 √(x + 5)² = √36 Square root both sides x + 5 = ± 6 x + 5 = 6 x + 5 = - 6 Solve for x - 5 - 5 - 5 - 5 ____________ _______________ x = 1 x = - 11

Factoring

Example: x² - 7x - 3 = - 2x - 9 Bring all terms to the left side + 2x + 9 + 2x + 9 ____________________________ x² - 5x + 6 = 0 Set the equation equal to zero Look at c (the constant) List its factors 1*6 & 2*3 Determine which factors add to b (x - 2)(x - 3) = 0 Put those factors in parentheses Determine the signs x - 2 = 0 x - 3 = 0 Solve for x + 2 + 2 + 3 + 3 ____________ ______________ x = 2 x = 3

Graphically

Look at the x-intercepts. If the graph does not intersect the x-axis, the roots are imaginary

What's the common factor in all of these methods?

They all start with setting the equation equal to zero.

Solution by Square Root

When b equals 0 Example: y = x² - 16 = 0 Make y equal 0 + 16 + 16 Add 16 to both sides ___________________ √x² = √16 Square root both sides x = ± 4

Quadratic Formula

x = -b ± √(b² - 4ac)/2a b² - 4ac is called the discriminant Example: x² + 10x - 11 x = - 10 ± √(10² - 4(1)(- 11) / 2(1) Substitute in x = - 10 ± √144 / 2 Square and distribute x = - 10 ± 12 / 2 Square root x = - 10 + 12 / 2 x = - 10 - 12 / 2 Solve for x x = 2 /2 x = -22 / 2 Combine like terms x = 1 x = - 11 Divide