Question $11 x^{2}-34 x+28=$

Studdy Solution

STEP 1

1. The given expression is a quadratic equation.
2. The goal is to factor the quadratic expression or solve for $x$ .
3. The quadratic equation can be factored if it has rational roots.

STEP 2

1. Identify the quadratic equation's coefficients $a$ , $b$ , and $c$ .
2. Use the quadratic formula or factorization to solve for $x$ .
3. Simplify the solutions for $x$ .

STEP 3

Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $11x^2 - 34x + 28$ . $a = 11, \quad b = -34, \quad c = 28$

STEP 4

To factor the quadratic equation, we need to find two numbers that multiply to $a \cdot c = 11 \cdot 28 = 308$ and add to $b = -34$ . We look for pairs of numbers that multiply to 308 and sum to -34.

STEP 5

Finding the pair: The numbers $-22$ and $-14$ multiply to 308 and add to -34.

STEP 6

Rewrite the middle term of the quadratic equation using the pair found: $11x^2 - 22x - 14x + 28$

STEP 7

Group the terms to factor by grouping: $(11x^2 - 22x) + (-14x + 28)$

STEP 8

Factor out the common factors in each group: $11x(x - 2) - 14(x - 2)$

STEP 9

Factor out the common binomial factor $(x - 2)$ : $(x - 2)(11x - 14)$

STEP 10

The factored form of the quadratic equation $11x^2 - 34x + 28$ is: $(x - 2)(11x - 14) = 0$

STEP 11

Set each factor to zero and solve for $x$ : $x - 2 = 0 \quad \text{or} \quad 11x - 14 = 0$

STEP 12

Solve the first equation: $x = 2$

STEP 13

Solve the second equation: $11x = 14$ $x = \frac{14}{11}$

STEP 14

The solutions to the quadratic equation $11x^2 - 34x + 28 = 0$ are: $x = 2 \quad \text{and} \quad x = \frac{14}{11}$

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