olve each of the following equations. State the exact and approximate answers as

Question

applicable. Use the solution method of your choice.
a) 4𝑥^2 = 4𝑥 + 15

Answer 1

To solve the equation 4𝑥^2 = 4𝑥 + 15, we can rearrange it and set it equal to zero:

4𝑥^2 - 4𝑥 - 15 = 0

Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use factoring:

First, we can divide the whole equation by 4 to simplify it:

𝑥^2 - 𝑥 - 15/4 = 0

To factor the equation, we want to find two binomials that multiply to give us 𝑥^2 - 𝑥 - 15/4. We need to find two numbers whose product is -15/4 and whose sum is -1 (since the coefficient of the 𝑥 term is -1). The numbers -4 and 15/4 satisfy these conditions:

(𝑥 - 4)(𝑥 + 15/4) = 0

Now we can set each factor equal to zero and solve for 𝑥:

𝑥 - 4 = 0 --> 𝑥 = 4

𝑥 + 15/4 = 0 --> 𝑥 = -15/4

So the exact solutions to the equation are 𝑥 = 4 and 𝑥 = -15/4.