Question

question 20, 4.2.47 the given triangle has area 35. find x.

Explanation:

Step1: Recall triangle - area formula

The area formula for a triangle is $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is the height. Here, $b = x + 3$ and $h=x$, and $A = 35$. So we have the equation $\frac{1}{2}(x)(x + 3)=35$.

Step2: Multiply both sides by 2

Multiply both sides of the equation $\frac{1}{2}(x)(x + 3)=35$ by 2 to get $x(x + 3)=70$. Expand the left - hand side: $x^{2}+3x=70$.

Step3: Rearrange to quadratic form

Rearrange the equation $x^{2}+3x=70$ to the standard quadratic form $ax^{2}+bx + c = 0$. So, $x^{2}+3x-70=0$.

Step4: Factor the quadratic equation

Factor the quadratic equation $x^{2}+3x - 70=0$. We need two numbers that multiply to $-70$ and add up to $3$. The numbers are $10$ and $-7$. So, $(x + 10)(x - 7)=0$.

Step5: Solve for x

Set each factor equal to zero: $x+10 = 0$ gives $x=-10$, and $x - 7=0$ gives $x = 7$. Since $x$ represents a length (height in the context of the triangle), we discard the negative solution.

Answer:

$7$