Calculator UseThis calculator is a quadratic equation solver that will solve a second-order polynomial equation in the form ax2 + bx + c = 0 for x, where a ≠ 0, using the completing the square method. Show
The calculator solution will show work to solve a quadratic equation by completing the square to solve the entered equation for real and complex roots. Completing the square when a is not 1To complete the square when a is greater than 1 or less than 1 but not equal to 0, factor out the value of a from all other terms. For example, find the solution by completing the square for: \( 2x^2 - 12x + 7 = 0 \) \( a \ne 1, a = 2 \) so divide through by 2 \( \dfrac{2}{2}x^2 - \dfrac{12}{2}x + \dfrac{7}{2} = \dfrac{0}{2} \) which gives us \( x^2 - 6x + \dfrac{7}{2} = 0 \) Now, continue to solve this quadratic equation by completing the square method. Completing the square when b = 0When you do not have an x term because b is 0, you will have a easier equation to solve and only need to solve for the squared term. For example: Solution by completing the square for: \( x^2 + 0x - 4 = 0 \) Eliminate b term with 0 to get: \( x^2 - 4 = 0 \) Keep \( x \) terms on the left and move the constant to the right side by adding it on both sides \( x^2 = 4\) Take the square root of both sides \( x = \pm \sqrt[]{4} \) therefore \( x = + 2 \) \( x = - 2 \) Calculator UseThis 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 formulaExample 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
square root equation calculator
How do you solve quadratic equations with square roots?To solve quadratic equations by the square root method, isolate the squared term and the constant term on opposite sides of the equation. Then take the square root of both sides, making the side with the constant term plus or minus the square root.
What are the 4 ways methods in solving the roots of a quadratic equation?The four methods of solving a quadratic equation are factoring, using the square roots, completing the square and the quadratic formula.
What is the square √ 64?The square root of 64 is 8, i.e. √64 = 8. The radical representation of the square root of 64 is √64. Also, we know that the square of 8 is 64, i.e. 82 = 8 × 8 = 64. Thus, the square root of 64 can also be expressed as √64 = √(8)2 = √(8 × 8) = 8.
|
Solving quadratic equations with square roots calculator
Related Posts
Copyright © 2024 ko.wiewird Inc.
