To solve the quadratic equation [tex]\( 4x^2 - 6 = 5x \)[/tex], follow these steps:
1. Rearrange the equation into the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
4x^2 - 5x - 6 = 0
\][/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -6 \)[/tex].
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the quadratic equation:
[tex]\[
a = 4, \quad b = -5, \quad c = -6
\][/tex]
3. Calculate the discriminant [tex]\(\Delta\)[/tex] using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we get:
[tex]\[
\Delta = (-5)^2 - 4 \cdot 4 \cdot (-6) = 25 + 96 = 121
\][/tex]
4. Determine the number of solutions based on the discriminant:
Since the discriminant [tex]\(\Delta = 121\)[/tex] is positive, there are two real solutions.
5. Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex] to find the solutions:
[tex]\[
x = \frac{-(-5) \pm \sqrt{121}}{2 \cdot 4}
\][/tex]
Simplifying this, we get:
[tex]\[
x = \frac{5 \pm 11}{8}
\][/tex]
6. Calculate the two solutions:
- For the positive root:
[tex]\[
x_1 = \frac{5 + 11}{8} = \frac{16}{8} = 2
\][/tex]
- For the negative root:
[tex]\[
x_2 = \frac{5 - 11}{8} = \frac{-6}{8} = -\frac{3}{4}
\][/tex]
Therefore, the solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = 2, -\frac{3}{4}
\][/tex]
