Quadratic Equations — Practice

Question 1

Solve: $x^2-8x+15=0$.

Accepted Answer

Two numbers multiplying to 15 and adding to 8: 3 and 5. $(x-3)(x-5)=0$, so $x=3$ or $x=5$.

Answer: x = 3 or 5.

Question 2

Solve: $y^2-3y-10=0$.

Accepted Answer

Two numbers multiplying to $-10$ and adding to $3$: 5 and $-2$. $(y-5)(y+2)=0$, so $y=5$ or $y=-2$.

Answer: y = 5 or -2.

Question 3

Solve $x^2-14x+48=0$ and verify the roots using the sum-and-product check.

Accepted Answer

Two numbers multiplying to 48 and adding to 14: 6 and 8. $(x-6)(x-8)=0$, so $x=6$ or $x=8$. Check: sum $=14$ (indeed $6+8=14$), product $=48$ (indeed $6	imes8=48$).

Answer: x = 6 or 8, verified.

Question 4

Solve $x^2-19x+90=0$ and $y^2-11y+30=0$. Compare $x$ and $y$.

Accepted Answer

Equation 1: $(x-9)(x-10)=0$, so $x=9$ or $x=10$. Equation 2: $(y-5)(y-6)=0$, so $y=5$ or $y=6$. Every combination gives $x>y$, since the smallest $x$ (9) still exceeds the largest $y$ (6).

Answer: x > y.

Question 5

Solve $x^2-11x+30=0$ and $y^2-17y+72=0$. Compare $x$ and $y$.

Accepted Answer

Equation 1: $(x-5)(x-6)=0$, so $x=5$ or $x=6$. Equation 2: $(y-8)(y-9)=0$, so $y=8$ or $y=9$. Every combination gives $x<y$, since the largest $x$ (6) is still less than the smallest $y$ (8).

Answer: x < y.

Question 6

Solve $x^2-x-12=0$ and $y^2-y-20=0$. Compare $x$ and $y$.

Accepted Answer

Equation 1: $(x-4)(x+3)=0$, so $x=4$ or $x=-3$. Equation 2: $(y-5)(y+4)=0$, so $y=5$ or $y=-4$. Checking all 4 combinations: $(4,5) o xy$; $(-3,5) o xy$ — both $x>y$ and $x

Question 7

Solve $x^2-12x+35=0$ and $y^2-13y+40=0$. Compare $x$ and $y$.

Accepted Answer

Equation 1: $(x-5)(x-7)=0$, so $x=5$ or $x=7$. Equation 2: $(y-5)(y-8)=0$, so $y=5$ or $y=8$. Checking all combinations: $(5,5) o x=y$; $(5,8) o xy$; $(7,8) o xy$ and $x

Question 8

Solve $x^2-8x+16=0$ and $y^2-9y+14=0$. Compare $x$ and $y$.

Accepted Answer

Equation 1: $(x-4)^2=0$, giving a single repeated root $x=4$. Equation 2: $(y-2)(y-7)=0$, so $y=2$ or $y=7$. Checking both combinations: $(4,2)	o x>y$; $(4,7)	o x<y$ — both directions occur.

Answer: Relationship cannot be determined.

Practice — Quadratic Equations

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