Question

Find the real solutions, if any, of each equation. Use any method. $$ 6 x^{2}+7 x-20=0 $$

Short answer

The real solutions are \( x = \frac{4}{3} \) and \( x = -\frac{5}{2} \).

Step-by-step solution

- Identify coefficients

Identify the coefficients in the quadratic equation. In the given equation, \(6x^2 + 7x - 20 = 0\), the coefficients are: \ a = 6, \ b = 7, \ c = -20.

- Calculate the discriminant

Use the formula for the discriminant, \( \Delta = b^2 - 4ac \). Substitute the values of a, b, and c: \ \( \Delta = 7^2 - 4(6)(-20) \) \ \( \Delta = 49 + 480 \) \ \( \Delta = 529 \). Since the discriminant is positive, there are two real solutions.

- Apply the quadratic formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \) to find the solutions. Substitute the values of a, b, and the discriminant: \ \( x = \frac{-7 \pm \sqrt{529}}{2 \times 6} \)

- Simplify the expression

Simplify the expression inside the square root and complete the calculations: \ \( x = \frac{-7 \pm 23}{12} \). This gives two solutions: \ \( x = \frac{-7 + 23}{12} = \frac{16}{12} = \frac{4}{3} \) \ \( x = \frac{-7 - 23}{12} = \frac{-30}{12} = -\frac{5}{2} \)

Key concepts

Discriminant

To solve a quadratic equation, the discriminant is an important value to consider. The discriminant helps us understand how many real solutions we can expect for the equation. The formula for the discriminant, denoted as \(\Delta\), is given by \(\Delta = b^{2} - 4ac\).
Here, 'a', 'b', and 'c' are the coefficients from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).
In our example, we identified these coefficients as:
- \(a = 6\)
- \(b = 7\)
- \(c = -20\)
Now, substituting the coefficients in the discriminant formula, we find:
\(\begin{align*} \Delta &= 7^2 - 4(6)(-20)\ \Delta &= 49 + 480\ \Delta &= 529.\ \end{align*}\)
A positive discriminant (\(\Delta > 0\)) indicates two real solutions. A zero discriminant (\(\Delta = 0\)) would indicate exactly one real solution, whereas a negative discriminant (\(\Delta < 0\)) indicates no real solutions.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. This formula is:
\(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).
Using the quadratic formula, you substitute the values of 'a', 'b', and the discriminant \( \Delta \).
For our equation \(6x^2 + 7x - 20 = 0\), we do the following:
- Substitute the values of 'a', 'b', and \( \Delta \) = 529:
\(\begin{align*} x &= \frac{-7 \pm \sqrt{529}}{2(6)}\ \frac{-7 \pm 23}{12} \end{align*}\)
This results in two expressions by considering both the positive and negative signs of 'plus/minus' (\(\pm\)).
- \(x = \frac{-7 + 23}{12}\)
- \(x = \frac{-7 - 23}{12}\).
This way, we use the quadratic formula to directly find the solutions.

Real Solutions

When solving a quadratic equation, finding real solutions is often the goal.
The real solutions are the x-values where the quadratic equation \(ax^2 + bx + c = 0\) equals zero.
Using the quadratic formula from the previous section, we further simplify:
\( \begin{align*} x &= \frac{-7 + 23}{12} = \frac{16}{12} = \frac{4}{3} \end{align*}\)
This gives us one real solution.
For the alternate solution:
\( \begin{align*} x &= \frac{-7 - 23}{12} = \frac{-30}{12} = -\frac{5}{2} \end{align*}\)
And this yields the second real solution.
Therefore, the equation \(6x^2 + 7x - 20 = 0\) has two real solutions:
- \(x = \frac{4}{3}\)
- \(x = -\frac{5}{2}\)
Understanding the discriminant and the quadratic formula allows us to confidently solve quadratic equations and determine the number of real solutions.