olve each of the following equations. State the exact and approximate answers as
applicable. Use the solution method of your choice.
a) 4π₯^2 = 4π₯ + 15
To solve the equation 4π₯^2 = 4π₯ + 15, we can rearrange it and set it equal to zero:
4π₯^2 - 4π₯ - 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:
π₯^2 - π₯ - 15/4 = 0
To factor the equation, we want to find two binomials that multiply to give us π₯^2 - π₯ - 15/4. We need to find two numbers whose product is -15/4 and whose sum is -1 (since the coefficient of the π₯ term is -1). The numbers -4 and 15/4 satisfy these conditions:
(π₯ - 4)(π₯ + 15/4) = 0
Now we can set each factor equal to zero and solve for π₯:
π₯ - 4 = 0 --> π₯ = 4
π₯ + 15/4 = 0 --> π₯ = -15/4
So the exact solutions to the equation are π₯ = 4 and π₯ = -15/4.