Quadratic Formula Calculator – Online Calculator

What do you mean by \(b^2 – 4ac < 0\) in a Quadratic Formula? To find the answer with all related tricky things, you may go through this Quadratic Formula Calculator article.

First of all, you just focus on the definition of Quadratic Formula.

What is the Quadratic Formula?

The quadratic formula is a solution to the quadratic equation which is any equation that can be rearranged in the form:

\[
ax^2 + bx + c = 0 \text{ where } a \neq 0
\]

Here, 

x is the variable you have to solve. 

a and b are coefficients. 

c is the constant.

The Formula:

From the quadratic formula, you can find the value of x as- 

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

This formula directly finds the roots of any quadratic equation.

For example: \( 3x^2 + 3x – 2 = 0 \)

Substituting the coefficients, you will find the roots to be-

\[
x = \frac{-3 \pm \sqrt{3^2 – 4 \cdot 3 \cdot (-2)}}{2 \cdot 3}
\]

This simplifies to two real and distinct roots. 

Components of quadratic formula

The term under the square root \(b^2-4ac\) is known as the discriminant. It is termed as “D”. It expresses the following: 

D > 0 provides two distinct real roots.

Consider the quadratic equation: 2x² – 4x – 6 = 0

Here, 

\[
\begin{align*}
a &= 2 \\
b &= -4, \\
c &= -6 \\
D &= b^2 – 4ac = (-4)^2 – 4 \cdot 2 \cdot (-6) = 16 + 48 = 64 \\
x &= \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} = \frac{4 \pm 8}{4} \\
x_1 &= \frac{4 + 8}{4} = \frac{12}{4} = 3 \\
x_2 &= \frac{4 – 8}{4} = \frac{-4}{4} = -1
\end{align*}
\]

D = 0 denotes exactly one real root (or, the two real roots are equal).

Consider the quadratic equation x² − 6x + 9 = 0.

Here,

\[
\begin{align*}
a &= 1 \\
b &= -6 \\
c &= 9 \\
D &= b^2 – 4ac = (-6)^2 – 4 \cdot 1 \cdot 9 = 36 – 36 = 0 \\
x &= \frac{-(-6) \pm \sqrt{0}}{2 \cdot 1} = \frac{6}{2} = 3
\end{align*}
\]

D < 0 shows that there are no real roots, but two complex roots.

Consider the quadratic equation x² + 2x + 5 = 0.

Here,

\[
\begin{align*}
a &= 1 \\
b &= 2 \\
c &= 5 \\
D &= b^2 – 4ac = (2)^2 – 4 \cdot 1 \cdot 5 = 4 – 20 = -16 \\
x &= \frac{-2 \pm \sqrt{-16}}{2 \cdot 1} = \frac{-2 \pm 4i}{2} = -1 \pm 2i
\end{align*}
\]

Properties of the Quadratic Formula

1. Symmetry: The roots may be symmetric with respect to the vertical axis of the parabola which is represented by the quadratic equation. This axis is located at \(x = – \frac{b}{2a}\).
2. Vertex Form: The quadratic formula helps find the apex of the parabola \(y=ax^2+bx+c\). The x-coordinate of the vertex is given by \(x= – \frac{b}{2a}​\) and its y-coordinate can be found by replacing this x into the original equation.
3. Axis of Symmetry: The line \(x= – \frac {b}{2a}\)​ ​acts as the axis of symmetry for the parabola and directly descends from the structure of the quadratic formula.
4. Dependence on Coefficients: The roots are directly influenced by the coefficients: a and b along with constant c. Changes in these values can affect the parabola as well as the nature of its roots.
5. Real and Complex Roots: The quadratic formula can have roots that are both real and complex. 
6. Sum and Product of Roots: The sum and product of the roots of a quadratic equation are as follows:
   • The sum: \( – \frac{b}{a}\)
   • The product: \(\frac{c}{a}\)

Because of complexity of quadratic formula, the Quadratic Formula Calculator has appeared as a mathematical assistant. It simplifies the solution process. The quadratic formula calculator with steps to follow is not just as a solver but an educational tool. 

Step by Step Guideline to Use Quadratic Formula Calculator

Using a quadratic formula calculator is straightforward. It can significantly simplify the process of cracking quadratic equations. The following shows an example and some tips and tricks for efficient use of the calculator.

1. Write the value of “Coefficient a”: This is the coefficient before x².
2. Write the value of “Coefficient b”: This is the coefficient before x.
3. Write the value of “Constant c”: This is the term which has no variable.
4. Click CALCULATE.

The calculator will display the roots which could be real or complex numbers reliant on the discriminant \(b^2 – 4ac\).

Example: Consider the quadratic equation: \(3x^2 -4x+6=0\)

Here, 

1. a = 3
2. b = −4, and 
3. c = −6.
4. Write a = 2 in the “Coefficient a”.
5. Write b = −4 in the “Coefficient b”.
6. Write c = −6 in the “Constant c”.
7. Click “CALCULATE”.

The calculator will then use the quadratic formula

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

to find the roots of the equation.
If the discriminant \((b^2-4ac)\) is less than zero (D < 0), an imaginary number arises. This means that the equation has not any real roots. It has only two complex roots. For your surprise, complex roots occur in conjugate pairs in the form:

\[
\begin{align*}
&a + bi \\
&\text{and} \\
&a − bi
\end{align*}
\]

\(
\begin{align*}
&\text{a and b are real numbers. } \\
& i \text{ is the imaginary unit. } \\
& i \text{ is defined as } \sqrt{-1}
\end{align*}
\)

Tips and Tricks for Efficient Use of the quadratic formula calculator

1. Before finding your coefficients, ensure the equation is in the standard form \(ax^2 +bx + c = 0\). This can prevent input errors.
2. If you’re entering negative values, make sure to use parentheses where necessary.
3. Knowing the role of the discriminant \((b^2-4ac)\) can help you anticipate the nature of the roots (real or complex) before even using the calculator.
4. Double-check your coefficients before clicking calculate to avoid rework.

By following these steps and tips, you can efficiently and accurately solve any quadratic equation.

Benefits of using quadratic formula calculator 

The need of the quadratic formula calculator with steps can’t be ignored. Here are the key points:

1. The quadratic formula calculator provides a reliable method for solving any quadratic equation. Its ability to find solutions consistently highlights its indispensability in the study of algebra.

2. The quadratic formula calculator acts as an important time-saver. It provides solutions quickly to quadratic equations that might take much longer to solve manually.

3. The quadratic formula calculator with steps breaks down the solving process into clear steps and enhances comprehension and retention of the method.

4. The quadratic formula step by step calculator allows students to probe deeper into the logic behind the quadratic formula. This not only helps in solving equations but also in understanding the mathematical principles.

5. Facilitated by the calculator, the mastery of the quadratic formula lays the basis for attempting more advanced topics in mathematics.

Final Thought

To sum up, the quadratic formula states a clear and concise method for solving quadratic equations. The introduction of quadratic formula calculator has made this solution process handy and plausible. If you’re a student dealing with algebra or an inquisitive mind to explore mathematical problems, the quadratic formula and its calculator counterparts would open up a world of analytical possibilities.

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