ICO Calculator with Quadratic Formula with 10 Solved Examples
ICO Calculator with Quadratic Formula
Decimal Roots: x₁ = Waiting…, x₂ = Waiting…
Fraction Roots: x₁ = –, x₂ = –
Mixed Fraction Roots: x₁ = –, x₂ = –
Algebra Quadratic equations are expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It is called “quadratic” because the highest power of the variable is two, which creates a parabolic curve when graphed.
Quadratic equations are widely used in real-life problem solving, such as calculating heights, speeds, profits, and projectile motion, and more often produce two solutions, known as roots, which are found using the quadratic formula to indicate the points where the parabola intersects the x-axis.
How to solve Quadratic Equation in Calculator
Type your valid equation into the input box exactly as it appears, using the on-screen keyboard for symbols like x² or √. of ICO Calculator with Quadratic formula, an online tool free.Ensure it includes an = sign, such as “2x² – 5x + 3 = 0,” so that it can correctly read both sides & extract math coefficients
When you enter the equation, the ICO calculator with Quadratic formula automatically formats expressions by converting x² into x**2 and inserting multiplication symbols where needed. This ensures the equation is interpreted correctly and prepares it for the solving process without requiring manual adjustments.
After entering the equation, press the Solve button. ICO calculator with Quadratic Formula instantly identifies the values of a, b, and c, then applies the quadratic formula. It calculates the discriminant, checks if roots are real or complex, and generates accurate decimal results for both x₁ and x₂.
Once the decimal roots are found, the ICO calculator With Quadratic formula converts them into simplified fractions and mixed-fraction formats. This helps users understand exact math values, not just rounded forms. All results update automatically and appear neatly in the results section.
This ICO calculator with Quadratic formula displays every step: identifying coefficients, computing the discriminant, applying the quadratic formula, substituting values, and reaching final answers. This helps users learn the method clearly by showing how each stage contributes to solving the equation.
Note: “ICO Calculator With Quadratic formula solves standard quadratic equations, but advanced users can also explore integer quadratic programming, a method where solutions are restricted to integers, useful in optimization problems in engineering and finance.”
List of Calculator Solving Quadratic Equations
10 Quadratic Equations for Practice
Examples of 10 Quadratic Equations with solutions
Quadratic Equations: Step-by-Step Solutions
Example 1: Method \(3x^2 – 11x + 8 = 0\)
Step 1: Identify coefficients:
a = 3, b = -11, c = 8
Step 2: formula for Quadratic equation:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{-(-11) \pm \sqrt{(-11)^2 – 4 \cdot 3 \cdot 8}}{2 \cdot 3} = \frac{11 \pm \sqrt{121 – 96}}{6} = \frac{11 \pm \sqrt{25}}{6} = \frac{11 \pm 5}{6} \]
Step 4: Final solutions:
\[ \begin{cases} x_1 = \frac{11 + 5}{6} = \frac{16}{6} = \frac{8}{3} \\[2mm] x_2 = \frac{11 – 5}{6} = \frac{6}{6} = 1 \end{cases} \]
Final Answer:
\[ x = \frac{8}{3} \quad \text{or} \quad x = 1 \]
Example 2: Method \(4x^2 + 7x – 15 = 0\)
Step 1: Identify the coefficients:
a = 4, b = 7, c = -15
Step 2: Applying formula Quadratic :
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{-7 \pm \sqrt{7^2 – 4 \cdot 4 \cdot (-15)}}{2 \cdot 4} = \frac{-7 \pm \sqrt{49 + 240}}{8} = \frac{-7 \pm \sqrt{289}}{8} = \frac{-7 \pm 17}{8} \]
Step 4: Final Ans:
\[ \begin{cases} x_1 = \frac{-7 + 17}{8} = \frac{10}{8} = \frac{5}{4} \\[2mm] x_2 = \frac{-7 – 17}{8} = \frac{-24}{8} = -3 \end{cases} \]
Final Answer:
\[ x = \frac{5}{4} \quad \text{or} \quad x = -3 \]
Example 3: Method \(5x^2 – 2x – 3 = 0\)
Step 1: Identification of coefficients:
a = 5, b = -2, c = -3
Step 2: Apply the formula:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 – 4 \cdot 5 \cdot (-3)}}{2 \cdot 5} = \frac{2 \pm \sqrt{4 + 60}}{10} = \frac{2 \pm \sqrt{64}}{10} = \frac{2 \pm 8}{10} \]
Step 4: Answer :
\[ \begin{cases} x_1 = \frac{2 + 8}{10} = 1 \\[2mm] x_2 = \frac{2 – 8}{10} = -\frac{3}{5} \end{cases} \]
Final Answer:
\[ x = 1 \quad \text{or} \quad x = -\frac{3}{5} \]
Example 4: Solution method \(2x^2 + 9x – 4 = 0\)
Step 1: Identify algebraic coefficients:
a = 2, b = 9, c = -4
Step 2: Applying Quadratic-formula:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{-9 \pm \sqrt{9^2 – 4 \cdot 2 \cdot (-4)}}{4} = \frac{-9 \pm \sqrt{81 + 32}}{4} = \frac{-9 \pm \sqrt{113}}{4} \]
Step 4: Final solutions:
\[ \begin{cases} x_1 = \frac{-9 + \sqrt{113}}{4} \\[2mm] x_2 = \frac{-9 – \sqrt{113}}{4} \end{cases} \]
Final Answer:
\[ x = \frac{-9 + \sqrt{113}}{4} \quad \text{or} \quad x = \frac{-9 – \sqrt{113}}{4} \]
Example 5: With Soltion \(6x^2 – 13x + 5 = 0\)
Step 1: look for coefficients in equation :
a = 6, b = -13, c = 5
Step 2: Aplly formula-Quadratic :
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{13 \pm \sqrt{169 – 120}}{12} = \frac{13 \pm \sqrt{49}}{12} = \frac{13 \pm 7}{12} \]
Step 4: Final solutions:
\[ \begin{cases} x_1 = \frac{13 + 7}{12} = \frac{20}{12} = \frac{5}{3} \\[2mm] x_2 = \frac{13 – 7}{12} = \frac{6}{12} = \frac{1}{2} \end{cases} \]
Final Answer:
\[ x = \frac{5}{3} \quad \text{or} \quad x = \frac{1}{2} \]
Example 6: Solution of equation \(7x^2 – 4x – 6 = 0\)
Step 1: Identify co-efficients:
a = 7, b = -4, c = -6
Step 2: Apply formula in this step :
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{4 \pm \sqrt{(-4)^2 – 4 \cdot 7 \cdot (-6)}}{14} = \frac{4 \pm \sqrt{16 + 168}}{14} = \frac{4 \pm \sqrt{184}}{14} = \frac{4 \pm 2\sqrt{46}}{14} = \frac{2 \pm \sqrt{46}}{7} \]
Step 4: Final result :
\[ \begin{cases} x_1 = \frac{2 + \sqrt{46}}{7} \\[2mm] x_2 = \frac{2 – \sqrt{46}}{7} \end{cases} \]
Final Answer:
\[ x = \frac{2 + \sqrt{46}}{7} \quad \text{or} \quad x = \frac{2 – \sqrt{46}}{7} \]
Example 7: Solve this equation \(8x^2 + 3x – 10 = 0\)
Step 1: Take look for coefficients:
a = 8, b = 3, c = -10
Step 2: Quadratic-formula application here :
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{-3 \pm \sqrt{3^2 – 4 \cdot 8 \cdot (-10)}}{16} = \frac{-3 \pm \sqrt{9 + 320}}{16} = \frac{-3 \pm \sqrt{329}}{16} \]
Step 4: Final Ans :
\[ \begin{cases} x_1 = \frac{-3 + \sqrt{329}}{16} \\[2mm] x_2 = \frac{-3 – \sqrt{329}}{16} \end{cases} \]
Final Answer:
\[ x = \frac{-3 + \sqrt{329}}{16} \quad \text{or} \quad x = \frac{-3 – \sqrt{329}}{16} \]
Example 8: Solve \(9x^2 – 5x + 1 = 0\)
Step 1: Identify coefficients:
a = 9, b = -5, c = 1
Step 2: Using Quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{5 \pm \sqrt{(-5)^2 – 4 \cdot 9 \cdot 1}}{18} = \frac{5 \pm \sqrt{25 – 36}}{18} = \frac{5 \pm \sqrt{-11}}{18} = \frac{5 \pm i\sqrt{11}}{18} \]
Step 4: Final solutions:
\[ \begin{cases} x_1 = \frac{5 + i\sqrt{11}}{18} \\[2mm] x_2 = \frac{5 – i\sqrt{11}}{18} \end{cases} \]
Final Answer:
\[ x = \frac{5 + i\sqrt{11}}{18} \quad \text{or} \quad x = \frac{5 – i\sqrt{11}}{18} \]
Example 9: Solve \(10x^2 + 11x – 12 = 0\)
Step 1: Identify coefficients:
a = 10, b = 11, c = -12
Step 2: Quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{-11 \pm \sqrt{11^2 – 4 \cdot 10 \cdot (-12)}}{20} = \frac{-11 \pm \sqrt{121 + 480}}{20} = \frac{-11 \pm \sqrt{601}}{20} \]
Step 4: Final solutions:
\[ \begin{cases} x_1 = \frac{-11 + \sqrt{601}}{20} \\[2mm] x_2 = \frac{-11 – \sqrt{601}}{20} \end{cases} \]
Final Answer:
\[ x = \frac{-11 + \sqrt{601}}{20} \quad \text{or} \quad x = \frac{-11 – \sqrt{601}}{20} \]
Example 10: Solve \(12x^2 – 17x + 6 = 0\)
Step 1: Identify coefficients:
a = 12, b = -17, c = 6
Step 2: Quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Step 3: Substitute the values:
\[ x = \frac{17 \pm \sqrt{(-17)^2 – 4 \cdot 12 \cdot 6}}{24} = \frac{17 \pm \sqrt{289 – 288}}{24} = \frac{17 \pm \sqrt{1}}{24} = \frac{17 \pm 1}{24} \]
Step 4: Final solutions:
\[ \begin{cases} x_1 = \frac{17 + 1}{24} = \frac{18}{24} = \frac{3}{4} \\[2mm] x_2 = \frac{17 – 1}{24} = \frac{16}{24} = \frac{2}{3} \end{cases} \]
Final Answer:
\[ x = \frac{3}{4} \quad \text{or} \quad x = \frac{2}{3} \]
Final Words
To master quadratic equations, consistent practice is essential. Use the ICO Calculator for Quadratic equations for verifying your answers, explore different equation forms, and understand each step of the quadratic formula.
Go through all the solved examples above, study the method of identifying coefficients, substituting values, and simplifying square roots, and then try similar problems on your own. Practicing these operations repeatedly will strengthen your concepts and improve accuracy.
Apply the same step-by-step approach shown in the examples, use the ICO calculator with Quadratic formula for verification, and develop confidence. With regular practice, you will quickly become skilled and efficient in solving quadratic equations.