Sure, let's solve the quadratic equation [tex]\( x^2 - 10x = 24 \)[/tex] step-by-step:
1. Rewrite the equation in standard form:
A quadratic equation in standard form is written as [tex]\( ax^2 + bx + c = 0 \)[/tex].
By subtracting 24 from both sides, the given equation becomes:
[tex]\[
x^2 - 10x - 24 = 0
\][/tex]
2. Identify the coefficients:
Here, the coefficients are:
[tex]\[
a = 1, \quad b = -10, \quad c = -24
\][/tex]
3. Calculate the discriminant:
The discriminant ([tex]\( \Delta \)[/tex]) of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-10)^2 - 4 \times 1 \times (-24) = 100 + 96 = 196
\][/tex]
4. Check the discriminant:
Since the discriminant is positive, the quadratic equation has two distinct real roots.
5. Calculate the roots using the quadratic formula:
The roots of the quadratic equation can be found using the formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging in the values:
[tex]\[
x = \frac{-(-10) \pm \sqrt{196}}{2 \times 1} = \frac{10 \pm 14}{2}
\][/tex]
6. Find the two roots:
[tex]\[
x_1 = \frac{10 + 14}{2} = \frac{24}{2} = 12
\][/tex]
[tex]\[
x_2 = \frac{10 - 14}{2} = \frac{-4}{2} = -2
\][/tex]
So, the solutions to the quadratic equation [tex]\( x^2 - 10x = 24 \)[/tex] are [tex]\( x = 12 \)[/tex] and [tex]\( x = -2 \)[/tex].
