Quadratic Equation Calculator With Steps

A Worked example to illustrate how the solve calculator Works:

Quadratic equation solver With Step by step Solution

A quadratic equation is basically a mathematical expression with one variable and f the form

ax^2+bx+c=0 with a, b are any numbers, c is a constant and \neq 0 .

Geometrically, a quadratic equation represents a parabola. Likewise, a solution to the quadratic equation represents a set of coordinates where the parabola crosses the x axis. Alternatively, the solution to a quadratic are the values of x ay y=0.

Finding the zeros of a quadratic polynomial calculator

Let p(x)=ax^2+bx+ c, \neq0 be a given quadratic equation. A real number ‘k’is said to be a zero of the quadratic polynomial if and only if p(k)=0.

There are 3 classical ways of finding the roots or zeroes of a quadratic equation namely.

• Graphing
• Quadratic formula
• Completing square method
• Factoring method

A quadratic equation is an equation of degree 2. This means that the highest power of the unknown or the variable is 2. Generally a quadratic equation has two solutions.

From the above general example, quadratics can exist in various forms. For instance, the values of coefficients b, c can be zero. Thus the problem will reduce to the solving of the equation ax^2=0 where a=/0 0. It is easy to see that the solution to this problem is just zero. The graph of such a quadratic will be a parabola whose vertex is the origin in a typical 2 d graph. This is a trivial case where the solution is just zero.

Moving on swiftly to more complex cases. If c=0, then we have a slightly interesting case scenario, whereby we will be solving an equation of the form ax^2+bx=0. With some basic algebra, it is easy to see that x(ax+b)=0 , thus, we have two distinct solutions with one of them been zero.

A more interesting case arises, when b=0. In such a case, the polynomial reduces to ax^2= , and thus x=\sqrt{c} .

Our calculator helps you find the value of x, no matter the nature of your quadratic equation.

A more interesting case is when all the constants a,b,c are non-zero. Solving such a problem is a non-trivial matter. Luckily, we have some 3 classical methods to help you solve such an equation. Our free online algebra calculator helps you work out the solution of any quadratic equation.

As you have seen from the above, calculations that involve finding the square root means that you can get results with both real and imaginary parts. Our online algebra calculator also supports the imaginary numbers. Likewise, the calculator will also print both the real and the imaginary parts of the roots to your polynomial equation.

Quadratic Equation Solver with Steps

The best part of using our online calculator is that it shows you all the workings. It is the only algebra calculator that shows you a step by step solution. Our quadratic equation solver step by step will show you all the working alongside the necessary explanation. It is truly a perfect way to learn algebra. The calculator, not only helps you find the value of x, but it also shows you how to manipulate and solve quadratic equations online. You can learn how to solve quadratic equations online by first trying to solve the equations on your own and then comparing your results with those of the calculator.

Quadratic equation solver by factoring: algebra 2 factoring calculator

Factoring is one of the fundamental ways of solving a quadratic equation. The example below illustrates how to solve a quadratic equation be factoring. The next example is an automatically generated solution as done with our factorize quadratic equation solver.

This method requires you to reduce your polynomial equation into 2 factors. From our original example, it is possible to write it in the form (x+h)(x+k)=0 , where (x+h), (x+k) are the factors of ax^2+bx+c=0 respectively. Nonetheless, not all polynomials can be factored, and therefore it is necessary to undertake some tests on your original equations before you start factoring it out. As a rule of thumb, a quadratic equation is factorable if an only if it has a positive Discriminant which is a perfect square. The Discriminanant to a quadratic equation is the quantity B^2-4AC . A negative Discriminant, means that the polynomial is not factorable over some real coefficients h,k.

For example the quadratic equation 3x^2-5x-2=0 is factorable since its discriminant is 49, which is a perfect square.

If your polynomial equation has some unreal roots, then you better try other available solution methods. Luckily our calculator also supports, complex roots.

Complex quadratic equation solver

Complex roots are of the form a+bi , where b\ne0 and i= -1^2 . The quadratic formula calculator is the best option when you have such an equation.

Quadratic formula solver with work

The quadratic formula calculator makes use of the quadratic formula

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

With the formula, you can find the roots of any quadratic equation just by plugging the values a,b,c on the right position within the formula.

The formula maintains that any polynomial of degree two can be solved using the formula where a,b,c are the constants in the equation respectively. With this calculator, it is easy to find the roots of any equation without worries. Simply enter your math in the text area provided, Hit calculate button to find the solution.

The roots of quadratic equations can either be real, complex or zero. A complex root means that the solution has both the real and an imaginary part of the form a+bi where i^2=-1 . On the other hand, a real solution means that the roots are all real numbers.

The automatic quadratic equation solver lets you see all the steps and working alongside the roots to your polynomial.

completing the square formula with this calculator with steps

A quadratic equation of the form ax² + bx + c = 0 for x, where a \ne 0 can be solved using the completing the square formula with this calculator. It is an online calculator capable of completing the square of a quadratic equation with all the steps.

Naturally some quadratics can be factored into a perfect square. For example

X^2-2x+1=0 can be factored into its constituent factor polynomials of degree 1 as follows:

(x-1)^2=0

With the later form, it is easy to solve the equation simply by taking square roots on either side of the expression. Doing so reduces the LHS into a first degree polynomial which is solvable through factorization and simplification.

Nice quadratics (factorable equations) are not always the case. In fact, most quadratics cannot be factored into first degree polynomials. Nevertheless, we can force perfect squares into such equations simply by adding a constant.

For Example x^2+6x+7=0 is not a perfect square. Nevertheless, we can create a perfect square out of it simply by adding 2. The result will be (x+3)^2=2 which is easier to solve than the earlier form.

discriminant and completing Square

The discriminant of a quadratic is a constant D= (b^2-4ac) . The constant is useful in determining the nature of the solution to a quadratic even before we engage into solving it. If a quadratic equation has a positive discriminant, then it has two roots. If D=0 , then there is a possibility of one non-zero solution. Lastly, if D is negative, the equation will have complex roots. A complex root will have both the real and the imaginary part.

Quadratic formula calculator imaginary

A quadratic equation that has a negative discriminant will always have at least 1 complex root. This calculator will help you to solve quadratic equations using the complete square formula even when the roots are complex numbers.

Let’s illustrate this further using a live example

Example: 2x^2-6x-5=0

Step 1: We write the polynomial/equation into standard form with leading coefficient equal 1 (Divide all coefficients by 2​):

x^2-3x-\frac{5}{2}=0

Step 2: Introduce a new Constant:  -\frac{3}{2} ​​ on the LHS to form a Perfect Square

(x^2-3x+(-\frac{3}{2})^2)-\frac{5}{2}-(-\frac{3}{2})^2=0

Step 3: Simplify the expression

(x-\frac{3}{2})^2-\frac{19}{4}=0

Step 4: Take Square root on both Sides of the Equation & Simplify

x-\frac{3}{2}=\pm \sqrt{\frac{19}{4}}

Step 5: Solution

x=\frac{(3+\sqrt{19})}{2}   OR  x=\frac{(3-\sqrt{19})}{2}

Solve a Quadratic Equation by Factoring Calculator

Notice that in the example above we first divide through by 2 to reduce the expression into a standard form equation. With a standard equation, it is now possible to apply the completing square procedure to determine the solution.