Question

1. $x^2 - 12x - 8$

Explanation:

Step1: Analyze the quadratic expression

We have the quadratic expression \(x^2 - 12x - 8\). Wait, the "29)" and "x" with the box might be a miswriting or a partial factoring attempt. Let's assume we want to factor \(x^2-12x - 8\) or maybe there's a typo. Wait, maybe the original problem is to factor or divide? Wait, the "29)" could be a typo, maybe "2x" or something else. Alternatively, if we consider the quadratic formula for \(ax^2+bx + c\), here \(a = 1\), \(b=- 12\), \(c=-8\). The discriminant \(D=b^2 - 4ac=(-12)^2-4\times1\times(-8)=144 + 32=176\). Then roots are \(x=\frac{12\pm\sqrt{176}}{2}=\frac{12\pm4\sqrt{11}}{2}=6\pm2\sqrt{11}\). But if it's a factoring with integer coefficients, \(x^2-12x - 8\) doesn't factor with integer roots. Wait, maybe the original expression was \(x^2-12x + 32\) (a common quadratic), which factors as \((x - 4)(x - 8)\). But with the given \(x^2-12x - 8\), using quadratic formula:

Step2: Apply quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). For \(x^2-12x - 8\), \(a = 1\), \(b=-12\), \(c = - 8\).

So \(x=\frac{12\pm\sqrt{(-12)^2-4\times1\times(-8)}}{2\times1}=\frac{12\pm\sqrt{144 + 32}}{2}=\frac{12\pm\sqrt{176}}{2}=\frac{12\pm4\sqrt{11}}{2}=6\pm2\sqrt{11}\).

But if there was a typo and the expression is \(x^2-12x + 35\), it factors as \((x - 5)(x - 7)\). Or \(x^2-12x + 32=(x - 4)(x - 8)\). Given the unclear notation with "29)" and the box, maybe it's a partial factoring or a division problem that's miswritten. Assuming we proceed with the quadratic formula for \(x^2-12x - 8\):

Answer:

The roots of \(x^2 - 12x - 8 = 0\) are \(x = 6 + 2\sqrt{11}\) and \(x = 6 - 2\sqrt{11}\)