Quadratic Equation Calculator

Understanding Quadratic Equations and How to Solve Them

Quadratic equations are polynomial equations of the second degree, typically written in the form ( ax^2 + bx + c = 0 ).

Here, ( a ), ( b ), and ( c ) are coefficients, and ( x ) represents the variable. The general goal is to find the values of ( x ) that satisfy the equation, known as the solutions or roots of the quadratic equation.

How to Solve Quadratic Equations:

1. Quadratic Formula:

The most common method to solve a quadratic equation is by using the quadratic formula:

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

Here’s a step-by-step guide:

Identify Coefficients: For the equation ( ax^2 + bx + c = 0 ), identify the coefficients ( a ), ( b ), and ( c ).

Calculate the Discriminant: Compute the discriminant ( Delta = b^2 - 4ac ). The discriminant determines the nature of the roots:

Positive Discriminant: Two distinct real roots.

Zero Discriminant: One real root (also known as a repeated root).

Negative Discriminant:** No real roots (the solutions are complex).

2. Example Calculation:

Let’s solve the quadratic equation :( 2x^2 - 4x - 6 = 0 ).

Identify Coefficients: ( a = 2 ), ( b = -4 ), ( c = -6 ).

Calculate the Discriminant:

[Delta = (-4)^2 - 4 * 2 * (-6)]

[Delta = 16 + 48 = 64]

Apply the Quadratic Formula:[x = frac{-(-4) pm sqrt{64}}{2 * 2}]

[x = frac{4 pm 8}{4}]

First Root:

[ x = frac{4 + 8}{4} = 3]

Second Root:

[x = frac{4 - 8}{4} = -1]

So, the solutions are ( x = 3 ) and ( x = -1 ).

Other Methods

Factoring:

If the quadratic can be factored, solve by setting each factor to zero.

Completing the Square:

Rewrite the equation in the form ( (x - h)^2 = k ) and solve for ( x ).

Quadratic equations are fundamental in algebra and have applications in various fields, including physics, engineering, and finance. Using methods like the quadratic formula ensures accurate solutions and helps understand the nature of the roots. Whether you’re solving homework problems or applying these concepts to real-world scenarios, mastering quadratic equations is a key skill in mathematics.

Frequently Asked Questions (FAQ) About Quadratic Equations

1. What is the Standard Form of a Quadratic Equation?

The standard form of a quadratic equation is written as: ax² + bx + c = 0, where a, b, and c are constants, and x represents the variable. The coefficient a should not be equal to 0.

2. What is the Vertex Form of a Quadratic Equation?

The vertex form of a quadratic equation is written as: y = a(x – h)² + k, where (h, k) represents the vertex of the parabola, and a determines the direction and width of the parabola.

3. What is the Discriminant of a Quadratic Equation?

The discriminant of a quadratic equation is the part of the quadratic formula under the square root: b² – 4ac. It helps determine the number and type of solutions the quadratic equation has:

If the discriminant is positive, there are two real and distinct solutions.
If the discriminant is zero, there is one real and repeated solution.
If the discriminant is negative, there are no real solutions, but two complex solutions.

4. How to Derive the Quadratic Formula from the Standard Form?

To derive the quadratic formula from the standard form ax² + bx + c = 0, follow these steps:

Step 1: Divide both sides of the equation by a.
Step 2: Move the constant c/a to the right side.
Step 3: Complete the square by adding (b/2a)² to both sides.
Step 4: Factor the left side as (x + b/2a)² and simplify the right side.
Step 5: Take the square root of both sides.
Step 6: Solve for x by isolating it on one side.

This process gives the quadratic formula: x = [ -b ± √(b² - 4ac) ] / 2a.

5. What is the Discriminant of the Quadratic Equation 0 = –x² + 4x – 2?

For the quadratic equation 0 = -x² + 4x - 2, the discriminant is calculated using b² – 4ac:

Here, a = -1, b = 4, and c = -2.
Discriminant = 4² – 4(-1)(-2) = 16 – 8 = 8.

So, the discriminant is 8.

6. How to Factor a Quadratic Equation?

To factor a quadratic equation in the form ax² + bx + c = 0, follow these steps:

Step 1: Find two numbers that multiply to a*c and add up to b.
Step 2: Rewrite the middle term using these two numbers.
Step 3: Group the terms and factor each group.
Step 4: Factor out the common binomial factor.

For example, x² – 5x + 6 = 0 can be factored as (x – 2)(x – 3) = 0.

7. What Are the Solutions to the Quadratic Equation x² – 16 = 0?

To solve the equation x² – 16 = 0, follow these steps:

Step 1: Recognize it as a difference of squares: x² – 16 = (x – 4)(x + 4).
Step 2: Set each factor equal to zero: x – 4 = 0 and x + 4 = 0.
Step 3: Solve for x: x = 4 and x = -4.

So, the solutions are x = 4 and x = -4.

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