Solve each equation using completing the square. See Examples 3 a...

Question

Solve each equation using completing the square. See Examples 3 and 4. x^2 - 7x + 12 = 0

Accepted Answer

Welcome back. I am so glad you're here. We are asked to complete the square and solve the given equation. Our given equation is P squared plus eight P minus 33 equals zero. Our answer choices are answer choice A, the solution set three negative 11, answer choice B the solution set 3 11, answer trace C the solution set negative three negative and answer trace D the solution set negative 3 11, all right, to complete the square. The first thing we look for is that our term to the second degree. So that's P squared has a coefficient of one and it does. Yay, that's great. Next, we want to move our constant term to the right hand side of the equation. That's this negative 33. We want to move it over because it's negative, we can add it to both sides. So we'll add 33 to both sides. And then where that constant term was, we will leave a space. So we'll bring down P squared plus eight P and then leave a space equals zero plus 33 is positive 33 on the right hand side. And we do leave a space after the constant term on the right hand side as well. And we are going to now fill in those spaces and take our steps to complete the square. So to complete the square, we take our coefficient of the term to the first degree. And that's this P it's coefficient is eight. So we're going to take eight, we're going to divide it by two and then square it. So inside the parentheses, eight, divided by two, that's a positive four, we still need to square it. But I want you to take note of that number inside of the parentheses. We'll need that again shortly, but taking four squared, that will be a positive 16. And we add that 16 to both sides of the equal sign where we left those spaces. So now we're going to factor the left hand side P squared plus eight P plus 16. And we know by definition of completing the square that that's going to factor to be open parentheses. And then we have a P and then it's going to be adding that term that we took note of inside of the parentheses before squaring it. That's going to be P plus four, close parentheses and squared. And that's our left hand side of the equation. Now, on the right, we take 33 plus 16 and we're going to have a positive 49. And now it's just a matter of solving for P. So because that quantity of P plus four is squared. We're going to take the square rut to free the P from the squared. And what we do to one side, we do to the other as well. On the left, the square root of the quantity of P plus four squared is just P plus four. And on the right, when we take the square root that's going to be a plus or minus. And the square root of 49 is a positive seven or a negative seven. So we've got P plus four equals plus or minus seven to get P by itself. We just need to subtract four from both sides of the equation. On the left four minus four is nothing. We just have P by itself. Yay. And on the right, we have a negative four plus or minus seven. So we have two different equations here. We have P equals negative four plus seven and P equals negative four minus seven. Now, we can just do the addition and subtraction negative four plus seven is a positive three. So if P equals three, a negative four minus seven is a negative 11. So P also equals negative 11. We look at our answer choices for the solution set of three, negative 11 and that matches with answer choice. A well done. We'll catch you on the next one.

Solve each equation using completing the square. See Examples 3 and 4. x^2 - 7x + 12 = 0

Key Concepts

Completing the Square

Quadratic Equations

Perfect Square Trinomial

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