Question

1. ( x^2 - 8x + 15 = 0 ) 6. ( x^2 - 6x + 8 = 0 )

Explanation:

Step1: Factor the quadratic equation

We need to find two numbers that multiply to \(15\) and add up to \(-8\) (the coefficient of \(x\) with a sign change). The numbers are \(-3\) and \(-5\) because \((-3)\times(-5) = 15\) and \((-3)+(-5)=-8\). So we can factor the quadratic as:
\(x^{2}-8x + 15=(x - 3)(x - 5)\)
So the equation becomes \((x - 3)(x - 5)=0\)

Step2: Solve for \(x\)

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
For \((x - 3)(x - 5)=0\), we have two cases:
Case 1: \(x-3=0\), then \(x = 3\)
Case 2: \(x - 5=0\), then \(x=5\)

Answer:

\(x = 3\) or \(x = 5\)