Solve using the quadratic formula. $$ 6 x^{2}=-7 x+20 $$ What are a, $$ b $$, and $$ c $$ ? $$ b=\square $$ $$ c=\square $$ What is the discriminant? After using the quadratic formula, what are the values of x ? $$ x=\square $$ $$ x=\square $$

Question

UpStudy AI · Accepted Answer

To solve the quadratic equation $$ 6x^{2} = -7x + 20 $$ using the quadratic formula, follow these steps:

### 1. Rewrite the Equation in Standard Form

First, bring all terms to one side of the equation to obtain the standard quadratic form $$ ax^{2} + bx + c = 0 $$:

$$
6x^{2} + 7x - 20 = 0
$$

### 2. Identify the Coefficients $$ a $$, $$ b $$, and $$ c $$

From the standard form, you can identify the coefficients as follows:

$$
\begin{align*}
a &= 6 \
b &= 7 \
c &= -20 \
\end{align*}
$$

So,

$$
b = \boxed{7}
$$

$$
c = \boxed{-20}
$$

### 3. Calculate the Discriminant

The discriminant ($$ D $$) is given by the formula:

$$
D = b^{2} - 4ac
$$

Plugging in the values:

$$
D = 7^{2} - 4 \times 6 \times (-20) = 49 + 480 = 529
$$

So, the discriminant is:

$$
\boxed{529}
$$

### 4. Apply the Quadratic Formula

The quadratic formula to find the roots ($$ x $$) of the equation is:

$$
x = \frac{ -b \pm \sqrt{D} }{2a}
$$

Substitute the known values:

$$
x = \frac{ -7 \pm \sqrt{529} }{2 \times 6} = \frac{ -7 \pm 23 }{12}
$$

### 5. Find the Values of $$ x $$

Calculate the two possible solutions:

1. **First Solution:**

$$
x = \frac{ -7 + 23 }{12} = \frac{16}{12} = \frac{4}{3}
$$

$$
x = \boxed{\frac{4}{3}}
$$

2. **Second Solution:**

$$
x = \frac{ -7 - 23 }{12} = \frac{-30}{12} = -\frac{5}{2}
$$

$$
x = \boxed{-\frac{5}{2}}
$$

### **Summary of Results**

- **Coefficients:**
  - $$ a = 6 $$
  - $$ b = \boxed{7} $$
  - $$ c = \boxed{-20} $$

- **Discriminant:**
  - $$ D = \boxed{529} $$

- **Solutions:**
  - $$ x = \boxed{\frac{4}{3}} $$
  - $$ x = \boxed{-\frac{5}{2}} $$

These are the two values of $$ x $$ that satisfy the original quadratic equation.
$$ a = 6 $$, $$ b = 7 $$, $$ c = -20 $$, discriminant = 529, $$ x = \frac{4}{3} $$ or $$ x = -\frac{5}{2} $$.

Solve using the quadratic formula.

What are a, , and ?

What is the discriminant?
After using the quadratic formula, what are the values of x ?

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

Solve using the quadratic formula. What are a, , and ? What is the discriminant? After using the quadratic formula, what are the values of x ?