Sure, let's solve the given problem step-by-step:
We start with the equation representing the problem:
[tex]\[ x + x^2 = 12 \][/tex]
First, we convert it into a standard form quadratic equation. To do that, let's rearrange the terms:
[tex]\[ x^2 + x - 12 = 0 \][/tex]
Now, we need to solve the quadratic equation [tex]\( x^2 + x - 12 = 0 \)[/tex]. To solve quadratic equations, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients from our quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. From the given equation [tex]\( x^2 + x - 12 = 0 \)[/tex], we identify:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 1 \][/tex]
[tex]\[ c = -12 \][/tex]
Next, we calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values we found:
[tex]\[ \Delta = (1)^2 - 4(1)(-12) \][/tex]
[tex]\[ \Delta = 1 + 48 \][/tex]
[tex]\[ \Delta = 49 \][/tex]
With a positive discriminant, we know there are two real solutions. Now, we substitute the values into the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{49}}{2 \times 1} \][/tex]
[tex]\[ x = \frac{-1 \pm 7}{2} \][/tex]
We calculate the two possible solutions separately:
[tex]\[ x_1 = \frac{-1 + 7}{2} \][/tex]
[tex]\[ x_1 = \frac{6}{2} \][/tex]
[tex]\[ x_1 = 3 \][/tex]
[tex]\[ x_2 = \frac{-1 - 7}{2} \][/tex]
[tex]\[ x_2 = \frac{-8}{2} \][/tex]
[tex]\[ x_2 = -4 \][/tex]
Hence, the two possible values of [tex]\( x \)[/tex] are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -4 \][/tex]
