Question

For what values of x are the following expressions equal to each other. the binomials x^2 - 10x +31 and x+1

Answer 1

To find the values of x for which the expressions x^2 - 10x + 31 and x + 1 are equal to each other, we can set them equal to each other and solve for x.

Setting x^2 - 10x + 31 equal to x + 1:

x^2 - 10x + 31 = x + 1

Rearranging the equation to standard quadratic form:

x^2 - 10x + 30 = 0

Now, we can factorize the quadratic expression:

(x - 5)(x - 6) = 0

Setting each factor equal to zero and solving for x:

x - 5 = 0 or x - 6 = 0

x = 5 or x = 6

So, the values of x for which the expressions x^2 - 10x + 31 and x + 1 are equal to each other are x = 5 and x = 6.

Answer 2

x = 5 or x = 6

Explanation:

Given expressions:

`\begin{cases}x^2-10x+31\\x+1\end{cases}`

To find the values of x for which the given expressions are equal to each other, first set the expressions equal to each other and rearrange so that we have a quadratic equal to zero:

`\begin{aligned}x^2-10x+31&=x+1\\x^2-10x+31-x&=x+1-x\\x^2-11x+31&=1\\x^2-11x+31-1&=1-1\\x^2-11x+30&=0\end{aligned}`

Factor the quadratic:

`\begin{aligned}x^2-11x+30&=0\\x^2-6x-5x+30&=0\\x(x-6)-5(x-6)&=0\\(x-5)(x-6)&=0\end{aligned}`

Using the zero-product property, set each factor equal to zero and solve for x:

`\begin{aligned}x-5=0 \implies x&=5\\x-6=0 \implies x&=6\\\end{aligned}`

Therefore, the values of x that make the given expressions equal to each other are x = 5 or x = 6.