Solving Quadratic Equations Using the Quadratic Formula

Q7. Solve: $x^2 + 5x = -2$

Background

Topic: Quadratic Equations and the Quadratic Formula

This question tests your ability to solve a quadratic equation by first rewriting it in standard form and then applying the quadratic formula.

Key Terms and Formulas

• Quadratic Equation (Standard Form): $ax^2 + bx + c = 0$

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $b^2 - 4ac$ (determines the nature of the roots)

Step-by-Step Guidance

1. Rewrite the equation in standard form: $x^2 + 5x + 2 = 0$.

2. Identify the coefficients: $a = 1$, $b = 5$, $c = 2$.

3. Write the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

4. Substitute the values of $a$, $b$, and $c$ into the formula: $x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$.

5. Calculate the discriminant $b^2 - 4ac$ and set up the expression under the square root, but do not simplify further yet.

Try solving on your own before revealing the answer!

Final Answer:

$x = \frac{-5 \pm \sqrt{17}}{2}$

We used the quadratic formula and simplified the discriminant to get the final answer in simplest radical form.

Q8. Solve: $4x^2 - 6x - 1 = 0$

Background

Topic: Quadratic Equations and the Quadratic Formula

This question asks you to solve a quadratic equation with leading coefficient $a \neq 1$ using the quadratic formula.

Key Terms and Formulas

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Coefficients: $a = 4$, $b = -6$, $c = -1$

Step-by-Step Guidance

1. Identify the coefficients: $a = 4$, $b = -6$, $c = -1$.

2. Write the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

3. Substitute the values into the formula: $x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4}$.

4. Calculate the discriminant $(-6)^2 - 4 \cdot 4 \cdot (-1)$ and set up the expression under the square root.

5. Write the next step as $x = \frac{6 \pm \sqrt{...}}{8}$, but do not simplify further yet.

Try solving on your own before revealing the answer!

Final Answer:

$x = \frac{3 \pm \sqrt{13}}{4}$

After simplifying the discriminant and dividing by $2a$, we get the solution in simplest radical form.

Q9. Solve: $x^2 - x + 5 = 0$

Background

Topic: Quadratic Equations and the Quadratic Formula

This question involves solving a quadratic equation where the discriminant is negative, leading to complex solutions.

Key Terms and Formulas

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant: $b^2 - 4ac$ (if negative, solutions are complex)

Step-by-Step Guidance

1. Identify the coefficients: $a = 1$, $b = -1$, $c = 5$.

2. Write the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

3. Substitute the values: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$.

4. Calculate the discriminant $(-1)^2 - 4 \cdot 1 \cdot 5$ and set up the expression under the square root.

5. Write the next step as $x = \frac{1 \pm \sqrt{...}}{2}$, but do not simplify further yet.

Try solving on your own before revealing the answer!

Final Answer:

$x = \frac{1 \pm i\sqrt{19}}{2}$

The discriminant is negative, so the solutions are complex numbers involving $i = \sqrt{-1}$.

Quadratic Formula Reference

Background

This is the general formula for solving any quadratic equation of the form $ax^2 + bx + c = 0$.

Key Formula

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

How to Use

1. Identify $a$, $b$, and $c$ from your quadratic equation.

2. Plug them into the formula above.

3. Simplify under the square root (the discriminant).

4. Continue simplifying to find the solutions for $x$.