Question

67 solve the following equation using technology. if two solutions, separate answers with a comma. if no solution, type no solution. \\(\sqrt{5x + 36} = x + 8\\)

Explanation:

Step1: Square both sides

To eliminate the square root, square both sides of the equation $\sqrt{5x + 36}=x + 8$. This gives $(\sqrt{5x + 36})^2=(x + 8)^2$, which simplifies to $5x + 36=x^2+16x + 64$.

Step2: Rearrange into quadratic form

Rearrange the equation to standard quadratic form $ax^2+bx + c = 0$. Subtract $5x$ and $36$ from both sides: $x^2+16x + 64-5x - 36 = 0$, so $x^2+11x + 28 = 0$.

Step3: Factor the quadratic

Factor the quadratic equation. We need two numbers that multiply to $28$ and add to $11$. Those numbers are $4$ and $7$. So, $x^2+11x + 28=(x + 4)(x + 7)=0$.

Step4: Solve for x

Set each factor equal to zero: $x + 4 = 0$ gives $x=-4$; $x + 7 = 0$ gives $x=-7$.

Step5: Check solutions

Check each solution in the original equation.
- For $x=-4$: Left side $\sqrt{5(-4)+36}=\sqrt{-20 + 36}=\sqrt{16}=4$; Right side $-4 + 8 = 4$. So, $x=-4$ is a valid solution.
- For $x=-7$: Left side $\sqrt{5(-7)+36}=\sqrt{-35 + 36}=\sqrt{1}=1$; Right side $-7 + 8 = 1$. Wait, but wait, let's re - check. Wait, original equation is $\sqrt{5x + 36}=x + 8$. When $x=-7$, right side is $1$, left side is $1$. But wait, let's check the domain of the square root. The expression under the square root must be non - negative: $5x+36\geq0\Rightarrow x\geq-\frac{36}{5}=-7.2$. Both $x=-4$ and $x=-7$ are in the domain. But wait, when we squared the equation, we might have introduced extraneous solutions, but in this case, both seem to work? Wait, no, wait when $x=-7$: $\sqrt{5*(-7)+36}=\sqrt{-35 + 36}=\sqrt{1}=1$, and $x + 8=-7 + 8 = 1$. When $x=-4$: $\sqrt{5*(-4)+36}=\sqrt{16}=4$, and $x + 8=-4 + 8 = 4$. But wait, let's graph the functions $y=\sqrt{5x + 36}$ and $y=x + 8$. The square root function $y=\sqrt{5x + 36}$ has a domain $x\geq - 7.2$ and is increasing. The line $y=x + 8$ has a slope of $1$ and y - intercept $8$. The intersection points: we found $x=-4$ and $x=-7$. But wait, let's check with technology (like a graphing calculator). If we graph $y=\sqrt{5x + 36}-x - 8$, we can find the roots. Alternatively, using a calculator to solve the original equation, we see that when we plug in $x=-7$, $\sqrt{5*(-7)+36}=\sqrt{1}=1$, and $-7 + 8 = 1$, so it works. When we plug in $x=-4$, $\sqrt{5*(-4)+36}=\sqrt{16}=4$, and $-4 + 8 = 4$, so it works. Wait, but the problem says "using technology". Let's use a graphing approach. The function $f(x)=\sqrt{5x + 36}-x - 8$. We can find where $f(x)=0$. By calculating, we see that both $x=-4$ and $x=-7$ are solutions? Wait, no, wait I made a mistake in factoring? Wait, $x^2+11x + 28$: $11x$? Wait, $16x-5x = 11x$, $64 - 36 = 28$. Yes. Then factoring: $(x + 4)(x + 7)=x^2+11x + 28$. Correct. But when we check $x=-7$ in the original equation: $\sqrt{5*(-7)+36}=\sqrt{1}=1$, and $x + 8 = 1$. So it's valid. But wait, let's check with the original equation again. Wait, maybe I made a mistake in the sign. Wait, the original equation is $\sqrt{5x + 36}=x + 8$. The right - hand side $x + 8$ must be non - negative because the square root is non - negative. So $x+8\geq0\Rightarrow x\geq - 8$. Both $x=-4$ and $x=-7$ are greater than or equal to $-8$. So both are valid? But wait, when we graph $y=\sqrt{5x + 36}$ and $y=x + 8$, let's find the intersection. For $x=-7$: $y=\sqrt{5*(-7)+36}=1$, $y=-7 + 8 = 1$. For $x=-4$: $y=\sqrt{5*(-4)+36}=4$, $y=-4 + 8 = 4$. So both are solutions? But wait, the problem says "if two solutions, separate answers with a comma". But wait, let's check with the quadratic formula. The quadratic is $x^2+11x + 28 = 0$, discriminant $D=121-112 = 9$, so roots $x=\frac{-11\pm3}{2}$. So $x=\frac{-11 + 3}{2}=\frac{-8}{2}=-4$, $x=\frac{-11-3}{2}=\frac{-14}{2}=-7$. So both are solutions. Wait, but maybe the problem expects us to use technology, like a graphing calculator, to find the solutions. When we graph $y=\sqrt{5x + 36}$ and $y=x + 8$, we can see that they intersect at $x=-7$ and $x=-4$. But wait, let's check $x=-7$ again. The square root of $5*(-7)+36$ is square root of $1$, which is $1$, and $x + 8$ is $1$. So it's correct. $x=-4$: square root of $5*(-4)+36$ is square root of $16$, which is $4$, and $x + 8$ is $4$. So both are solutions. But wait, maybe I made a mistake in the problem statement? Wait, the original equation is $\sqrt{5x + 36}=x + 8$. Let's re - check the algebra. Squaring both sides: $5x + 36=x^2+16x + 64$. Then $x^2+11x + 28 = 0$. Factoring: $(x + 4)(x + 7)=0$. So solutions $x=-4$ and $x=-7$. Both satisfy the original equation. So the solutions are $-7,-4$.

Answer:

-7, -4