What are Roots of a Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form $$ax^2 + bx + c = 0$$ where $a \neq 0$. The roots of a quadratic equation are the values of $x$ that make the equation true.

Finding the Roots of a Quadratic Equation

There are several methods for finding the roots of a quadratic equation. One common method is the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

The Discriminant

The discriminant of a quadratic equation is the quantity $b^2 - 4ac$. The discriminant determines the number and nature of the roots of the equation:

• If $D > 0$, the equation has two distinct real roots.
• If $D = 0$, the equation has one repeated real root (also called a double root).
• If $D < 0$, the equation has no real roots (also called imaginary roots).

Examples

Here are some examples of finding the roots of quadratic equations:

• Example 1: Find the roots of the equation $x^2 - 4x + 3 = 0$.

Using the quadratic formula, we have: $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)}$$ $$x = \frac{4 \pm \sqrt{16 - 12}}{2}$$ $$x = \frac{4 \pm 2}{2}$$ $$x = 2 \pm 1$$

So the roots of the equation are $x = 1$ and $x = 3$.

• Example 2: Find the roots of the equation $2x^2 + 3x + 1 = 0$.

Using the quadratic formula, we have: $$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(1)}}{2(2)}$$ $$x = \frac{-3 \pm \sqrt{9 - 8}}{4}$$ $$x = \frac{-3 \pm 1}{4}$$ $$x = \frac{-2}{4} \quad \text{or} \quad x = \frac{-4}{4}$$ $$x = -\frac{1}{2} \quad \text{or} \quad x = -1$$

So the roots of the equation are $x = -\frac{1}{2}$ and $x = -1$.

• Example 3: Find the roots of the equation $x^2 + 2x + 5 = 0$.

Using the quadratic formula, we have: $$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(5)}}{2(1)}$$ $$x = \frac{-2 \pm \sqrt{4 - 20}}{2}$$ $$x = \frac{-2 \pm \sqrt{-16}}{2}$$ $$x = \frac{-2 \pm 4i}{2}$$ $$x = -1 \pm 2i$$

So the roots of the equation are $x = -1 + 2i$ and $x = -1 - 2i$.

Nature of the Roots

Discriminant

The discriminant of a quadratic equation $$ax^2 + bx + c = 0$$ is given by the formula $$D = b^2 - 4ac$$

The discriminant determines the nature of the roots of the quadratic equation:

• If $D > 0$, the equation has two distinct real roots.
• If $D = 0$, the equation has one repeated real root (also called a double root).
• If $D < 0$, the equation has no real roots (also called imaginary roots).

Examples

Example 1: Consider the quadratic equation $$x^2 - 4x + 3 = 0$$. The discriminant of this equation is $$D = (-4)^2 - 4(1)(3) = 16 - 12 = 4$$. Since $$D > 0$$, the equation has two distinct real roots.

Example 2: Consider the quadratic equation $x^2 - 6x + 9 = 0$. The discriminant of this equation is $D = (-6)^2 - 4(1)(9) = 36 - 36 = 0.$ Since $D = 0$, the equation has one repeated real root.

Example 3: Consider the quadratic equation $x^2 + 4x + 5 = 0$. The discriminant of this equation is $D = (4)^2 - 4(1)(5) = 16 - 20 = -4.$ Since $D < 0$, the equation has no real roots.

Conclusion

The discriminant of a quadratic equation determines the nature of its roots. By calculating the discriminant, we can determine whether the equation has two distinct real roots, one repeated real root, or no real roots.

Finding Roots of a Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are real numbers, and $a$ is non-zero.

The roots of a quadratic equation are the values of $x$ that make the equation true.

Solving a Quadratic Equation

There are several methods for solving a quadratic equation. One common method is the quadratic formula:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Using the Quadratic Formula

To use the quadratic formula, simply substitute the values of $a$, $b$, and $c$ into the formula and solve for $x$.

For example, to solve the quadratic equation $x^2 - 3x - 4 = 0$, we would substitute $a = 1$, $b = -3$, and $c = -4$ into the quadratic formula:

$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)}$$

Simplifying this expression, we get:

$$x = \frac{3 \pm \sqrt{9 + 16}}{2}$$

$$x = \frac{3 \pm \sqrt{25}}{2}$$

$$x = \frac{3 \pm 5}{2}$$

So the roots of the equation $x^2 - 3x - 4 = 0$ are $x = 4$ and $x = -1$.

Other Methods for Solving Quadratic Equations

In addition to the quadratic formula, there are several other methods for solving quadratic equations, including:

• Completing the square
• Factoring
• Using a graph

The method you choose will depend on the specific equation you are solving.

Quadratic equations are a common type of algebraic equation, and there are several methods for solving them. The quadratic formula is a versatile method that can be used to solve any quadratic equation.

Word Problems on Finding Roots of a Quadratic Equation

Quadratic equations are equations of the form $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants and $x$ is the variable. The roots of a quadratic equation are the values of $x$ that make the equation true.

Example 1: Finding the Roots of a Quadratic Equation by Factoring

Find the roots of the quadratic equation $x^2 - 5x - 6 = 0$

Solution:

1. Factor the quadratic equation:

$$x^2 - 5x - 6 = (x - 6)(x + 1) = 0$$

2. Set each factor equal to zero and solve for $x$:

$$x - 6 = 0 \quad \Rightarrow \quad x = 6$$

$$x + 1 = 0 \quad \Rightarrow \quad x = -1$$

Therefore, the roots of the quadratic equation are $x = 6$ and $x = -1$.

Example 2: Finding the Roots of a Quadratic Equation by Using the Quadratic Formula

Find the roots of the quadratic equation $$2x^2 + 3x - 5 = 0.$$

Solution:

1. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 2$, $b = 3$, and $c = -5$.

2. Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)}$$ $$x = \frac{-3 \pm \sqrt{9 + 40}}{4}$$ $$x = \frac{-3 \pm \sqrt{49}}{4}$$ $$x = \frac{-3 \pm 7}{4}$$

Therefore, the roots of the quadratic equation are $x = \frac{1}{2}$ and $x = -2$.

Applications of Quadratic Equations

Quadratic equations have many applications in real life, including:

• Finding the roots of a quadratic equation can be used to solve problems involving projectile motion.
• Finding the roots of a quadratic equation can be used to solve problems involving the area of a rectangle.
• Finding the roots of a quadratic equation can be used to solve problems involving the volume of a sphere.

Quadratic equations are a powerful tool that can be used to solve a variety of problems in real life. By understanding how to find the roots of a quadratic equation, you can open up a whole new world of possibilities.

Find Roots of Quadratic Equation FAQs

What is the quadratic formula?

The quadratic formula is a mathematical equation that can be used to find the roots of a quadratic equation. The quadratic formula is:

$$x = (-b ± \sqrt{(b² - 4ac)}) / 2a$$

where:

• x is the unknown variable
• a, b, and c are the coefficients of the quadratic equation

How do I use the quadratic formula?

To use the quadratic formula, you need to know the values of a, b, and c for the quadratic equation you are trying to solve. Once you have these values, you can plug them into the quadratic formula and solve for x.

What are the roots of a quadratic equation?

The roots of a quadratic equation are the values of x that make the equation equal to zero. The roots of a quadratic equation can be real numbers, imaginary numbers, or complex numbers.

What is the difference between real and imaginary roots?

Real roots are numbers that can be represented on the number line. Imaginary roots are numbers that cannot be represented on the number line. Imaginary roots are always multiples of the imaginary unit i, which is defined as the square root of -1.

What is a complex number?

A complex number is a number that has both a real part and an imaginary part. Complex numbers can be represented in the form a + bi, where a is the real part and b is the imaginary part.

How do I find the roots of a quadratic equation with complex roots?

To find the roots of a quadratic equation with complex roots, you can use the quadratic formula. When you plug the values of a, b, and c into the quadratic formula, you will get two complex roots. The complex roots will be conjugates of each other, which means that they will have the same real part but opposite imaginary parts.

What are some examples of quadratic equations?

Here are some examples of quadratic equations:

• x² + 2x + 1 = 0
• 2x² - 3x - 5 = 0
• -x² + 4x - 7 = 0

How can I solve a quadratic equation without using the quadratic formula?

There are a few ways to solve a quadratic equation without using the quadratic formula. One way is to use the factoring method. The factoring method involves factoring the quadratic equation into two linear factors. Once the quadratic equation is factored, you can set each linear factor equal to zero and solve for x.

Another way to solve a quadratic equation without using the quadratic formula is to use the completing the square method. The completing the square method involves adding and subtracting a constant term to the quadratic equation so that it can be written in the form of a perfect square. Once the quadratic equation is in the form of a perfect square, you can take the square root of both sides of the equation and solve for x.