Question

extra credit (up to 5 pts.) after many years of trying, mr. howard created the perfect angle. however, before mr. howard could measure the size of his angle, dr. moore came and bisected it. with mr. howard feeling dejected and defeated, dr. moore gave him one hint: the two angles that he just created could be expressed by the algebraic expressions 5x² - 7x + 32 and 4x² - 19x + 45. what was the size of mr. howards original perfect angle?

Explanation:

Step1: Set the two angle - expressions equal

Since an angle is bisected, the two resulting angles are equal. So we set $5x^{2}-7x + 32=4x^{2}-19x + 45$.

Step2: Rearrange the equation

Move all terms to one side of the equation:
$5x^{2}-4x^{2}-7x + 19x+32 - 45=0$.
Simplify to get $x^{2}+12x - 13=0$.

Step3: Factor the quadratic equation

Factor $x^{2}+12x - 13$ as $(x + 13)(x - 1)=0$.

Step4: Solve for x

Set each factor equal to zero:
$x+13 = 0$ gives $x=-13$; $x - 1=0$ gives $x = 1$. Since angles are non - negative in this context, we use $x = 1$.

Step5: Find the measure of one of the bisected angles

Substitute $x = 1$ into either of the original angle - expressions. Let's use $5x^{2}-7x + 32$.
When $x = 1$, we have $5(1)^{2}-7(1)+32=5 - 7+32=30$.

Step6: Find the measure of the original angle

Since the original angle is twice the measure of the bisected angle, the original angle is $2\times30 = 60$.

Answer:

60