Solving Quadratics

$x = 3$

Rewrite as $5x^2 - 30x + 45 = 0$ and factor out 5: $5(x^2 - 6x + 9) = 0$. This is $5(x - 3)^2 = 0$, so $x = 3$.

$x = -3$: $x^2 - 6x + 9$ factors as $(x - 3)^2$, not $(x + 3)^2$.

$x = -5$: Factoring out 5 is only the first step; 5 is not a solution. Keep factoring $x^2 - 6x + 9$.

$x = 5$: Factoring out 5 is only the first step; 5 is not a solution. Keep factoring $x^2 - 6x + 9$.

2

Find two numbers that multiply to −22 and add to 9: 11 and −2. So $(x + 11)(x - 2) = 0$, giving $x = -11$ or $x = 2$. The positive solution is 2.

11: Set each factor equal to zero: $x + 11 = 0$ gives $x = -11$, not 11.

-11: That's the other solution. The question asks for the positive one.

4.5

Rewrite as $4x^2 + 12x - 135 = 0$. Quadratic formula: $x = \frac{-12 \pm \sqrt{144 + 2160}}{8} = \frac{-12 \pm 48}{8}$. The positive solution is $\frac{36}{8} = 4.5$.

-7.5: That's the negative solution. The question asks for the positive one.

$x = 1.2$

Factor: $(5x - 6)(x - 6) = 0$. So $5x - 6 = 0$ gives $x = 1.2$, or $x = 6$.

$x = -6$: This comes from $(5x + 6)(x + 6)$, which has a middle term of $+36x$. Check the middle term.

$x = 4$: This comes from $(5x - 9)(x - 4)$, which doesn't multiply back to the original. Check the middle term.

$x = -1.8$: This comes from $(5x + 9)(x + 4)$, which doesn't multiply back to the original. Check the middle term.

$x=-2$

Factor out 3: $3(x^2-10x-24)=3(x-12)(x+2)=0$. The solutions are 12 and −2; −2 is listed.

$x=-12$: The factor $x-12$ gives $x=12$, not −12.

$x=-4$: Substituting −4 does not make the expression zero.

$x=-6$: Use factors of −24 whose sum is −10: −12 and 2.

$\{-3,-2.5\}$

Move all terms to one side: $2x^2+11x+15=0$. Factor: $(2x+5)(x+3)=0$, so $x=-2.5$ or $x=-3$.

$\{-5,-1.5\}$: These values give $2x^2+13x+15$ when used to build the factors, not $2x^2+11x+15$.

$\{2.5,3\}$: Both solutions are negative because the factored expressions are $2x+5$ and $x+3$.

$\{1.5,5\}$: These are positive and do not satisfy the original equation.

$x = -3$ and $x = 2$

Take the square root of both sides: $2x + 1 = 5$ or $2x + 1 = -5$. Solving these equations gives $x = 2$ and $x = -3$.

$x \approx -4.65$ and $x \approx 0.65$

Using the quadratic formula gives $x = -2 \pm \sqrt{7}$. These values are approximately $-4.65$ and $0.65$.

9 ft

Set the equation equal to zero: $x^2 + 3x - 108 = 0$. Factor: $(x + 12)(x - 9) = 0$, so $x = -12$ or $x = 9$. A width must be positive, so the deck is 9 feet wide.

6 ft: −6 and 18 multiply to −108, but they do not add to 3, the linear coefficient.

4 ft: −4 and 27 multiply to −108, but they do not add to 3, the linear coefficient.

3 ft: Setting the factor $x + 3$ equal to zero ignores the 108. Move every term to one side before factoring.

–11 and 1

Take the square root of both sides: $x + 5 = \pm 6$. Subtract 5: $x = 6 - 5 = 1$ and $x = -6 - 5 = -11$.

–5 and 6: This solves $x + 5 = 0$ and $x^2 = 36$ separately, using only the positive root.

–1 and 11: The square root was correct, but 5 was added instead of subtracted.

–6 and 5: This solves $x + 5 = 0$ with a sign error and uses only the negative root of 36.