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Fundamentals
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1.5
Equations
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3. Solving Quadratic Equations

4. Solving Quadratic Equations
Quadratic equations are second-degree equations like x2 + 2x – 3 = 0 or 2x2 + 3 = 5x.

5. Solving Quadratic Equations
This means that we add a constant to an expression to make it a perfect square. For example, to make x2 – 6x a perfect square, we must add 9, since x2 – 6x + 9 = (x – 3)2.

6. Example 6 – Solving Quadratic Equations by Completing the Square
Solve each equation.
(a) x2 – 8x + 13 = (b) 3x2 – 12x + 6 = 0
Solution:
(a) x2 – 8x + 13 = 0
x2 – 8x = –13
x2 – 8x + 16 = –
Given equation
Subtract 13
Complete the square: add = 16

7. Example 6 – Solution
cont’d
(b) After subtracting 6 from each side of the equation, we must factor the coefficient of x2 (the 3) from the left side to put the equation in the correct form for completing the square.
Perfect square
Take square root
Add 4

8. Example 6 – Solution 3x2 – 12x + 6 = 0 3x2 – 12x = –6 3(x2 – 4x) = –6
cont’d
3x2 – 12x + 6 = 0
3x2 – 12x = –6
3(x2 – 4x) = –6
Now we complete the square by adding (–2)2 = inside the parentheses. Since everything inside the parentheses is multiplied by 3, this means that we are actually adding 3  4 = 12 to the left side of the equation.
Given equation
Subtract 6
Factor 3 from LHS

9. Example 6 – Solution Thus, we must add 12 to the right side as well.
cont’d
Thus, we must add 12 to the right side as well.
3(x2 – 4x + 4) = –6 + 3  4
3(x – 2)2 = 6
(x – 2)2 = 2
Complete the square: add 4
Perfect square
Divide by 3
Take square root
Add 2