How to find zeros of a quadratic function calculator

• Algebra Calculator
• Practice
• Lessons
• Pricing
• Log In
• Sign Up

Quadratic Formula Calculator

What do you want to calculate?

Example: 2x^2-5x-3=0

Step-By-Step Example

Learn step-by-step how to use the quadratic formula!

Example (Click to try)

2x2−5x−3=0

About the quadratic formula

Solve an equation of the form ax2+bx+c=0 by using the quadratic formula:

x=

Quadratic Formula Video Lesson

Need more problem types? Try MathPapa Algebra Calculator

EMBED

Make your selections below, then copy and paste the code below into your HTML source.
* For personal use only.

Theme

Output Type

Lightbox
Popup
Inline

Widget controls displayed
Widget results displayed

To add the widget to Blogger, click here and follow the easy directions provided by Blogger.

To add the widget to iGoogle, click here. On the next page click the "Add" button. You will then see the widget on your iGoogle account.

To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source:

For self-hosted WordPress blogs

To add a widget to a MediaWiki site, the wiki must have the Widgets Extension installed, as well as the code for the Wolfram|Alpha widget.

To include the widget in a wiki page, paste the code below into the page source.

Calculator Use

This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula.

The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant \( (b^2 - 4ac) \) is less than, greater than or equal to 0.

 When \( b^2 - 4ac = 0 \) there is one real root.

 When \( b^2 - 4ac > 0 \) there are two real roots.

 When \( b^2 - 4ac < 0 \) there are two complex roots.

Quadratic Formula:

The quadratic formula

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a } \)

is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2)

\( ax^2 + bx + c = 0 \)

Examples using the quadratic formula

Example 1: Find the Solution for \( x^2 + -8x + 5 = 0 \), where a = 1, b = -8 and c = 5, using the Quadratic Formula.

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a } \)

\( x = \dfrac{ -(-8) \pm \sqrt{(-8)^2 - 4(1)(5)}}{ 2(1) } \)

\( x = \dfrac{ 8 \pm \sqrt{64 - 20}}{ 2 } \)

\( x = \dfrac{ 8 \pm \sqrt{44}}{ 2 } \)

The discriminant \( b^2 - 4ac > 0 \) so, there are two real roots.

Simplify the Radical:

\( x = \dfrac{ 8 \pm 2\sqrt{11}\, }{ 2 } \)

\( x = \dfrac{ 8 }{ 2 } \pm \dfrac{2\sqrt{11}\, }{ 2 } \)

Simplify fractions and/or signs:

\( x = 4 \pm \sqrt{11}\, \)

which becomes

\( x = 7.31662 \)

\( x = 0.683375 \)

Example 2: Find the Solution for \( 5x^2 + 20x + 32 = 0 \), where a = 5, b = 20 and c = 32, using the Quadratic Formula.

\( x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a } \)

\( x = \dfrac{ -20 \pm \sqrt{20^2 - 4(5)(32)}}{ 2(5) } \)

\( x = \dfrac{ -20 \pm \sqrt{400 - 640}}{ 10 } \)

\( x = \dfrac{ -20 \pm \sqrt{-240}}{ 10 } \)

The discriminant \( b^2 - 4ac < 0 \) so, there are two complex roots.

Simplify the Radical:

\( x = \dfrac{ -20 \pm 4\sqrt{15}\, i}{ 10 } \)

\( x = \dfrac{ -20 }{ 10 } \pm \dfrac{4\sqrt{15}\, i}{ 10 } \)

Simplify fractions and/or signs:

\( x = -2 \pm \dfrac{ 2\sqrt{15}\, i}{ 5 } \)

which becomes

\( x = -2 + 1.54919 \, i \)

\( x = -2 - 1.54919 \, i \)

calculator updated to include full solution for real and complex roots

An online find real zeros calculator determines the zeros (exact, numerical, real, and complex) of the functions on the given interval. The find the zeros of the function calculator computes the linear, quadratic, polynomial, cubic, rational, irrational, quartic, exponential, hyperbolic, logarithmic, trigonometric, hyperbolic, and absolute value function. Read on to understand more about how to find zeros of a function. Let’s start with some basics!

What are Zeros of a Function?

In mathematics, the zeros of real numbers, complex numbers, or generally vector functions f are members x of the domain of ‘f’, so that f (x) disappears at x. The function (f) reaches 0 at the point x, or x is the solution of equation f (x) = 0.

Additionally, for a polynomial, there may be some variable values for which the polynomial will be zero. These values ​​are called polynomial zeros. They are sometimes called the roots of polynomials that could easily be determined by using this best find all zeros of the polynomial function calculator. We find the zeros or roots of a quadratic equation to find the solution of a given equation.

The standard form of the polynomial in x is \( a_nx^n + a_{n-1}x^{n-1} +….. + a_1x + a_0 \), where \( x_n, x_{n-1},… .., x_1, x_0 \) are constants, \( a_n ≠ 0 \), and n is an integer.

For example, algebraic expressions like \( \sqrt {a + a + 5, a^2 + 1 / a^2} \) are not polynomials, because all exponents of x in the expression are not integers. You can also verify the details by this free zeros of polynomial functions calculator.

Zeros Formula:

Assume that P (x) = 9x + 15 is a linear polynomial with one variable.

Let’s the value of ‘x’ be zero in P (x), then

\( P (x) = 9k + 15 = 0 \)

So, k \( = -15/9 = -5 / 3 \)

Generally, if ‘k’ is zero of the linear polynomial in one variable P(x) = mx + n, then

P(k) = mk + n = 0

k = – n / m

It can be written as,

Zero polynomial K = – (constant / coefficient (x))

The above expressions are also taken into consideration for calculations by best find the zeros calculator with steps.

However, an online Binomial Theorem Calculator helps you to find the expanding binomials for the given binomial equation.

How to Find the Zeros of a Function?

Find all real zeros of the function is as simple as isolating ‘x’ on one side of the equation or editing the expression multiple times to find all zeros of the equation. Generally, for a given function f (x), the zero point can be found by setting the function to zero. The x value that indicates the set of the given equation is the zeros of the function. To find the zero of the function, find the x value where f (x) = 0.

In simple words, the zero of a function can be defined as the point where the function becomes zeros. The degree of the function is the maximum degree of the variable x.

·        A function of degree 1 is called a linear function.

The standard form is ax + b,

Where, a and b are real numbers, and a ≠ 0.

7x + 23 is an example of linear function.

·         The function with degree 2 is called the quadratic function.

The standard form is \( ax^2 + bx + c \),

Where, a, b, and c are real numbers and a ≠ 0.

\( X^2 + 10x + 12 \) is an example of a quadratic function.

·         The degree 3 of a function is called the cubic function.

The standard form is \( ax^3 + bx^2 + cx + d \),

Where a, b, c, and d are real numbers, and a ≠ 0.

\( X^3 + 44x^2 + 23x + 2 \) is an example of a cubic function.

Similarly,

\( Y^6 + 23y^5 + 13y^4 + 32y^3 + 65y^2 + y + 22 \) is a function of degree 6 of y.

Key points:

·         All linear functions have only one zero.

·         The zero point of a function depends on its degree.

However, a handy Inflection Point Calculator to find points of inflection and concavity intervals of the given equation.

Example:

If the degree of the function is \( x^3 + m^{a-4} + x^2 + 1 \), is 10, what does value of ‘a’?

Solution:

The degree of the function P(m) is the maximum degree of m in P(m).

Therefore, the complex finding zeros calculator takes the \( m^{a-4} = m^4 \)

$$ a-4 = 10, a = 4 + 10 = 14 $$

Hence, the value of ‘a’ is 14.

Example:

Calculate the sum and zeros product of the quadratic function \( 4x^2 – 9 \).

Solution:

The quadratic function is \( 4x^2 – 9 \)

The complex zero calculator can be writing the \( 4x^2 – 9 \) value as \( 2.2x^2-(3.3) \)

Where, it is (2x + 3) (2x-3).

For finding zeros of a function, the real zero calculatorset the above expression to 0

$$ (2x + 3) (2x-3) = 0 $$

$$ 2x + 3 = 0 $$

$$ 2x = -3 $$

$$ X = -3/2 $$

Similarly, the zeros of a function calculator takes the second value 2x-3 = 0

$$ 2x = 3 $$

$$ x = 3/2 $$

So, zeros of the function are 3/2 and -3/2

Therefore, zeros finder take the Sum and product of the function:

Zero sum = \( (3/2) + (-3/2) = (3/2) – (3/2) = 0 \)

Zero product = \( (3/2). (-3/2) = -9/4 \).

How Find All Zeros Calculator Works?

An online polynomial zero calculator compute the zeros for several functions on the given interval by following these guidelines:

Input:

·         Enter an equation for finding zeros of a function.

·         Hit the calculate button to see the results.

Output:

·         The real polynomial zeros calculator with steps finds the exact and real values of zeros and provides the sum and product of all roots.

FAQ:

How do you find the roots of a polynomial?

The zeros of a polynomial calculator can find all zeros or solution of the polynomial equation P (x) = 0 by setting each factor to 0 and solving for x.

Are zeros and roots the same?

According to the rule of thumbs: zero refers to a function (such as a polynomial), and the root refers to an equation.

What does real zeros mean?

The real zero of the function is the real number when the function value is zero. If f (r) = 0, then the real number r is the root of the function f. If you wonder about how to find all zeros of a function, let this free real zero calculator assist you in this regard.

Conclusion:

Use this online find zeros calculator to find all zeros of the given expression. Find zeros can be time-consuming, there might be lots of possible roots and for each term, you should check whether or not it’s an actual zero(root). Fortunately, there is our zeros solver, which can do all these calculations for you quickly.

Reference:

From the source of Wikipedia: Zero of a function, Polynomial roots, Fundamental theorem of algebra, Zero set.

From the source of Study for Mathematics: find the zeros of a function, find zeros of a quadratic function, zeros of a polynomial function.

From the source of Lumen Learning: Find zeros of a polynomial function, Analysis of the Solution, FUNCTION WITH REPEATED REAL ZEROS.

What is the formula to find the zeros of a quadratic equation?