Question

Find the real solutions, if any, of each equation. Use any method. $$ 9 x^{2}-12 x+4=0 $$

Short answer

The real solution is \( x = \frac{2}{3} \).

Step-by-step solution

Identify the quadratic equation

The given quadratic equation is \[ 9x^{2} - 12x + 4 = 0 \] This is in the standard form of a quadratic equation, \[ ax^{2} + bx + c = 0 \], with \( a = 9 \), \( b = -12 \), \( c = 4 \).

Calculate the discriminant

The discriminant \( \triangle \) of a quadratic equation \( ax^{2} + bx + c = 0 \) is given by \[ \triangle = b^{2} - 4ac \]. Substituting the values of \( a \), \( b \), and \( c \): \[ \triangle = (-12)^{2} - 4 \times 9 \times 4 = 144 - 144 = 0 \].

Interpret the discriminant

Since the discriminant is zero (\( \triangle = 0 \)), the quadratic equation has exactly one real solution. This means the equation has a repeated root.

Find the root using the quadratic formula

The quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]. Since the discriminant is zero, the formula reduces to \[ x = \frac{-b}{2a} \]. Substitute \( b = -12 \) and \( a = 9 \): \[ x = \frac{12}{2 \times 9} = \frac{12}{18} = \frac{2}{3} \].

Key concepts

Discriminant

The discriminant is a key concept in solving quadratic equations. It is derived from the coefficients of the equation and provides essential information about the nature of the roots. The formula to calculate the discriminant is given by \[ \Delta = b^{2} - 4ac. \] This small term reveals whether the roots of the quadratic equation are real or complex.

If the discriminant is positive, the equation has two distinct real roots.
If the discriminant is zero, the equation has exactly one real root.
If the discriminant is negative, the equation has no real roots, but instead has two complex roots.

In our example, the discriminant was computed as follows: \[( -12 ) ^ {2} - 4 \times 9 \times 4 = 144 - 144 = 0 \] Because the discriminant is zero, we know there is exactly **one real solution**. This leads us into the next important concept.

Repeated Root

A repeated root, also known as a double root, occurs when the discriminant of a quadratic equation is zero. In such cases, both roots of the quadratic equation are the same. It means the parabola touches the x-axis at only one point.

Using our example equation \[ 9x ^ {2} - 12x + 4 = 0 \], we found that the discriminant is zero (\[ \Delta = 0 \]).

This results in one real, repeated root. To find this root, we can still use the quadratic formula which simplifies in this special case.

Quadratic Formula

The quadratic formula is a universal method used to solve any quadratic equation. The general formula is expressed as:
\[ x = \frac { -b \pm \sqrt { b ^ {2} - 4ac } } { 2a } \] This formula yields the solutions (roots) of the quadratic equation. However, when the discriminant (\