Problem 33
Question
\(x^{2}+4 x-9=0\)
Step-by-Step Solution
Verified Answer
The solutions are \(-2 + \sqrt{13}\) and \(-2 - \sqrt{13}\).
1Step 1: Identify the coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 1, b = 4, c = -9\).
2Step 2: Calculate the Discriminant
Use the discriminant formula \(D = b^2 - 4ac\) to find the discriminant. \(D = 4^2 - 4 \cdot 1 \cdot (-9) = 16 + 36 = 52\).
3Step 3: Apply the Quadratic Formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute the values: \(a = 1, b = 4, D = 52\).
4Step 4: Solve for the roots
Using the quadratic formula, \(x = \frac{-4 \pm \sqrt{52}}{2 \cdot 1} = \frac{-4 \pm 2 \sqrt{13}}{2} = -2 \pm \sqrt{13}\).
Key Concepts
quadratic formuladiscriminantroots of quadratic equationscoefficients in quadratic equations
quadratic formula
The **quadratic formula** is a powerful tool used to find the roots of any quadratic equation. Quadratic equations are in the form \(ax^2 + bx + c = 0\). To solve this, the quadratic formula is used: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula works for any values of \(a\), \(b\), and \(c\) as long as \(a \eq 0\).
In our exercise, we applied the quadratic formula to the equation \(x^2 + 4x - 9 = 0\). By substituting the identified coefficients \(a = 1\), \(b = 4\), and \(c = -9\), we were able to determine the roots efficiently.
In our exercise, we applied the quadratic formula to the equation \(x^2 + 4x - 9 = 0\). By substituting the identified coefficients \(a = 1\), \(b = 4\), and \(c = -9\), we were able to determine the roots efficiently.
discriminant
The **discriminant** is a component of the quadratic formula that determines the nature of the roots. The discriminant is given by the expression \(D = b^2 - 4ac\).
The value of the discriminant reveals important information:
In our specific case, the discriminant calculation was \(D = 4^2 - 4 \cdot 1 \cdot (-9) = 52\). Since \(52 > 0\), we confirmed that there are two distinct real roots for the equation \(x^2 + 4x - 9 = 0\).
The value of the discriminant reveals important information:
- **If D > 0**: There are two distinct real roots.
- **If D = 0**: There is exactly one real root.
- **If D < 0**: There are two complex roots.
In our specific case, the discriminant calculation was \(D = 4^2 - 4 \cdot 1 \cdot (-9) = 52\). Since \(52 > 0\), we confirmed that there are two distinct real roots for the equation \(x^2 + 4x - 9 = 0\).
roots of quadratic equations
Finding the **roots of quadratic equations** is essential to solving these types of problems. The roots are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Using the quadratic formula, we can solve for these roots.
For the quadratic equation \(x^2 + 4x - 9 = 0\), we substituted the values into the formula: \[ x = \frac{-4 \pm \sqrt{52}}{2 \cdot 1} = \frac{-4 \pm 2 \sqrt{13}}{2} \].
Simplifying, we found the roots to be \(-2 \pm \sqrt{13}\). These are the values of \(x\) that solve the original quadratic equation.
For the quadratic equation \(x^2 + 4x - 9 = 0\), we substituted the values into the formula: \[ x = \frac{-4 \pm \sqrt{52}}{2 \cdot 1} = \frac{-4 \pm 2 \sqrt{13}}{2} \].
Simplifying, we found the roots to be \(-2 \pm \sqrt{13}\). These are the values of \(x\) that solve the original quadratic equation.
coefficients in quadratic equations
The **coefficients in quadratic equations** play a crucial role in determining the behavior and solutions of the equation. In the equation \(ax^2 + bx + c = 0\),
- \