Answer:
[tex]p = -4[/tex] and [tex]q = 1[/tex]
Step-by-step explanation:
Given
[tex]x^2 + 8x + 15 = 0[/tex]
Required
Rewrite as:
[tex](x - p)^2 =q[/tex]
[tex]x^2 + 8x + 15 = 0[/tex]
Subtract 15 from both sides
[tex]x^2 + 8x + 15 - 15 = 0 - 15[/tex]
[tex]x^2 + 8x = - 15[/tex]
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To make the equation a perfect square, follow these steps
[tex]b = 8[/tex] ---- the coefficient of x
Divide both sides by 2:
[tex]\frac{b}{2} = \frac{8}{2}[/tex]
[tex]\frac{b}{2} = 4[/tex]
Square both sides
[tex](\frac{b}{2})^2 = 4^2[/tex]
[tex](\frac{b}{2})^2 = 16[/tex]
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So, we add 16 t0 both sides of: [tex]x^2 + 8x = - 15[/tex]
[tex]x^2 + 8x + 16 = - 15 + 16[/tex]
[tex]x^2 + 8x + 16 = 1[/tex]
Factorize:
[tex]x^2 + 4x + 4x+ 16 = 1[/tex]
[tex]x(x + 4) + 4(x + 4) = 1[/tex]
[tex](x + 4) (x + 4) = 1[/tex]
[tex](x + 4)^2 = 1[/tex]
By comparison to: [tex](x - p)^2 =q[/tex]
[tex]-p = 4[/tex] and [tex]q = 1[/tex]
So, we have:
[tex]p = -4[/tex] and [tex]q = 1[/tex]
