-7×² + 14× + 21=0 using quadractic formula

To solve the quadratic equation \(-7x^2 + 14x + 21 = 0\) using the quadratic formula, follow these steps:

### Step 1: Identify the Coefficients
In the standard quadratic equation \(ax^2 + bx + c = 0\), the coefficients are:

\[
a = -7, \quad b = 14, \quad c = 21
\]

### Step 2: Write the Quadratic Formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 3: Substitute the Coefficients into the Formula
Substituting \(a = -7\), \(b = 14\), and \(c = 21\) into the formula:

\[
x = \frac{-14 \pm \sqrt{14^2 - 4(-7)(21)}}{2(-7)}
\]

### Step 4: Simplify the Expression
First, calculate the discriminant (\(\Delta\)):

\[
\Delta = 14^2 - 4(-7)(21) = 196 + 588 = 784
\]

Now substitute the discriminant back into the quadratic formula:

\[
x = \frac{-14 \pm \sqrt{784}}{-14}
\]

Since \(\sqrt{784} = 28\), we have:

\[
x = \frac{-14 \pm 28}{-14}
\]

### Step 5: Solve for Both Values of \(x\)
Now, calculate the two possible values for \(x\):

1. \(x_1 = \frac{-14 + 28}{-14} = \frac{14}{-14} = -1\)
2. \(x_2 = \frac{-14 - 28}{-14} = \frac{-42}{-14} = 3\)

Thus, the solutions to the equation are \(x = -1\) and \(x = 3\).

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How do you derive the quadratic formula from a general quadratic equation?
2. What is the significance of the discriminant in determining the nature of the roots?
3. How do you solve a quadratic equation if the discriminant is negative?
4. What is the geometric interpretation of the solutions of a quadratic equation?
5. Can the quadratic formula be used for equations that are not quadratic?

**Tip:** Always simplify the coefficients of the quadratic equation if possible, as it can make calculations easier when applying the quadratic formula.

Learn how to solve the quadratic equation -7x^2 + 14x + 21 using the quadratic formula with detailed step-by-step instructions. This problem covers key algebraic concepts, including the quadratic formula and discriminant. Suitable for students in Grades 10-12.

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