Question $2 b^{2}=b+6$

Studdy Solution

STEP 1

1. The equation $2b^2 = b + 6$ is a quadratic equation in standard form.
2. Solving the equation will involve rearranging terms to set the equation to zero.
3. The solutions can be found using factoring, completing the square, or the quadratic formula.

STEP 2

1. Rearrange the equation to standard quadratic form.
2. Solve the quadratic equation using an appropriate method.
3. Verify the solutions.

STEP 3

Rearrange the equation to bring all terms to one side, setting the equation to zero:
$2b^2 = b + 6$
Subtract $b$ and $6$ from both sides:
$2b^2 - b - 6 = 0$

STEP 4

Solve the quadratic equation. We will attempt to factor the quadratic expression:
$2b^2 - b - 6 = 0$
Look for two numbers that multiply to $2 \times (-6) = -12$ and add to $-1$ . These numbers are $-4$ and $3$ .
Rewrite the middle term using these numbers:
$2b^2 - 4b + 3b - 6 = 0$

STEP 5

Factor by grouping. Group the terms and factor out the common factors:
$(2b^2 - 4b) + (3b - 6) = 0$
Factor out the greatest common factor from each group:
$2b(b - 2) + 3(b - 2) = 0$

STEP 6

Factor out the common binomial factor:
$(2b + 3)(b - 2) = 0$

STEP 7

Use the zero product property to find the solutions. Set each factor equal to zero:
$2b + 3 = 0 \quad \text{or} \quad b - 2 = 0$
Solve for $b$ :
$2b + 3 = 0 \quad \Rightarrow \quad 2b = -3 \quad \Rightarrow \quad b = -\frac{3}{2}$
$b - 2 = 0 \quad \Rightarrow \quad b = 2$
The solutions to the equation are:
$b = -\frac{3}{2} \quad \text{and} \quad b = 2$

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