Question

the width of a yard is four less than a number. the length is five more than twice that same number. the area of the yard is 15 square meters. write and solve a quadratic equation to find the value of x, then choose the answer that best describes the value of x. a x = 5 and x = -3.5 are both solutions b x = 5 is the only solution c x = -3.5 is the only solution d neither x = 5 nor x = -3.5 are solutions

Explanation:

Step1: Set up the area formula

The area of a rectangle is $A = length\times width$. Given length $l=2x + 5$, width $w=x - 4$ and $A = 15$. So, $(2x + 5)(x - 4)=15$.

Step2: Expand the left - hand side

Using the FOIL method, $(2x+5)(x - 4)=2x\times x-2x\times4+5\times x - 5\times4=2x^{2}-8x + 5x-20=2x^{2}-3x-20$. So the equation becomes $2x^{2}-3x-20 = 15$.

Step3: Rearrange to standard quadratic form

Subtract 15 from both sides: $2x^{2}-3x-20 - 15=0$, which simplifies to $2x^{2}-3x-35 = 0$.

Step4: Factor the quadratic equation

We need to find two numbers that multiply to $2\times(-35)=-70$ and add up to - 3. The numbers are - 10 and 7. So, $2x^{2}-3x - 35=2x^{2}-10x+7x - 35=2x(x - 5)+7(x - 5)=(2x + 7)(x - 5)=0$.

Step5: Solve for x

Set each factor equal to zero:
If $2x+7 = 0$, then $2x=-7$, and $x=-\frac{7}{2}=-3.5$.
If $x - 5=0$, then $x = 5$.
But since the width $x - 4$ and length $2x + 5$ represent real - world dimensions (lengths), the width $x-4>0$ (a negative width doesn't make sense in this context). When $x=-3.5$, the width $x - 4=-3.5 - 4=-7.5<0$. When $x = 5$, the width $x - 4=5 - 4 = 1$ and the length $2x+5=2\times5+5 = 15$.

Answer:

B. $x = 5$ is the only solution