Quadratic Equation Root Calculation

This solution finds the second root of the quadratic equation $x^2 - 19x + 88 = 0$, given that one root is 8.

Equation and Known Root

The provided quadratic equation is:

$x^2 - 19x + 88 = 0$

We are given that one root, let's call it $\alpha$, is 8.

Finding the Other Root Using Sum of Roots

In a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots ($\alpha + \beta$) is given by the formula $-b/a$.

• For the equation $x^2 - 19x + 88 = 0$, we have $a=1$, $b=-19$, and $c=88$.
• The sum of the roots is $\alpha + \beta = -(-19)/1 = 19$.
• Since we know $\alpha = 8$, we can substitute it into the sum: $8 + \beta = 19$.
• Solving for $\beta$, we get: $\beta = 19 - 8 = 11$.

Alternative Method: Product of Roots

The product of the roots ($\alpha \beta$) for the equation $ax^2 + bx + c = 0$ is given by $c/a$.

• The product of the roots is $\alpha \beta = 88/1 = 88$.
• Substituting the known root $\alpha = 8$: $8 \beta = 88$.
• Solving for $\beta$: $\beta = 88 / 8 = 11$.

Result

Both methods show that the other root ($\beta$) of the equation $x^2 - 19x + 88 = 0$ is 11.