Question

the total area of the figure to the right is 434 cm². use this fact to write an equation involving x. then solve the equation to find the value of x. the equation is □.

Explanation:

Step1: Analyze the figure's area composition

The figure can be seen as a large rectangle. The area of a rectangle is length times width. The total area of the figure is given as 434 $cm^{2}$. The length of the figure is $(x + 6)$ cm and the width is $(x+23)$ cm. So the equation based on the area - formula $A = lw$ is $(x + 6)(x + 23)=434$.

Step2: Expand the left - hand side

Using the FOIL method, $(x + 6)(x + 23)=x^{2}+23x+6x + 138=x^{2}+29x + 138$. So the equation becomes $x^{2}+29x+138 = 434$.

Step3: Rearrange to standard quadratic form

Subtract 434 from both sides of the equation: $x^{2}+29x+138−434 = 0$, which simplifies to $x^{2}+29x - 296 = 0$.

Step4: Solve the quadratic equation

We can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for a quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 1$, $b = 29$, and $c=-296$. First, calculate the discriminant $\Delta=b^{2}-4ac=(29)^{2}-4\times1\times(-296)=841 + 1184=2025$. Then, $x=\frac{-29\pm\sqrt{2025}}{2}=\frac{-29\pm45}{2}$. We have two solutions: $x_1=\frac{-29 + 45}{2}=\frac{16}{2}=8$ and $x_2=\frac{-29 - 45}{2}=\frac{-74}{2}=-37$. Since length cannot be negative in this context, we discard $x=-37$.

Answer:

$x = 8$