4D - The general quadratic formula
When quadratic equations become more complex, factorisation and using the null factor law to solve becomes a long and arduous process. However, the general quadratic formula can often be efficiently applied to solve these problems.
■ To find the solution(s) to the quadratic equation \(ax^2+bx+c=0\) the general quadratic formula states \[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\]
To use the formula to solve quadratic equations, simply substitute the coefficients that correspond to \(a\), \(b\) and \(c\) in the general quadratic equation (\(ax^2+bx+c\)).
4D - Example 1: Solving quadratic equations using the general quadratic formula
Use the general quadratic formula to solve the following equation for \(x\): \[x^2+4x+1=0\]
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4D - Example 1: Video solution
4D - Example 1: Practice
Question 1: Using the general quadratic formula, solve the equation \(x^2-6x+7=0\) for \(x\). Question 2: Using the general quadratic formula, solve the equation \(-x^2-8x-12=0\) for \(x\). 4D - Example 1: Solutions
Question 1: \[x=3-\sqrt{2},x=3+\sqrt{2}\] Question 2: \[x=-6,x=-2\] |
4D - Example 2: Solving quadratic equations using the general quadratic formula
Use the general quadratic formula to solve the following equation for \(x\): \[x^2-5x-20=0\]
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4D - Example 2: Video solution
4D - Example 2: Practice
Question 1: Using the general quadratic formula, solve the equation \(x^2-11x+24=0\) for \(x\). Question 2: Using the general quadratic formula, solve the equation \(x^2+10x+19=0\) for \(x\). 4D - Example 2: Solutions
Question 1: \[x=3,x=8\] Question 2: \[x=-5-\sqrt{6},x=-5+\sqrt{6}\] |
4D - Example 3: Solving quadratic equations using the general quadratic formula
Use the general quadratic formula to solve the following equation for \(x\): \[3x^2+x-12=0\]
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4D - Example 3: Video solution
4D - Example 3: Practice
Question 1: Using the general quadratic formula, solve the equation \(3x^2+10x-8=0\) for \(x\). Question 2: Using the general quadratic formula, solve the equation \(2x^2+4x=8\) for \(x\). 4D - Example 3: Solutions
Question 1: \[x=-4, x=\frac{2}{3}\] Question 2: \[x=-1-\sqrt{5},x=-1+\sqrt{5}\] |
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Additional Exercises
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Topic Worksheets
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Other Resources
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Solve each of the following equations for \(x\)
(a) \(x^2-5x-6=0\) (b) \(2x^2-4x+7=0\) Solve the equation \(9x^2-9x+2=0\) for \(x\).
Solve the equation \(3x^2+5x=7\) for \(x\).
Solve each of the following equations for \(x\)
(a) \(x^2-6x+2=0\) (b) \(3x^2+5x+1=0\) (c) \(-x^2+8x=-3\) |
Solutions
Question 1: (a) \(x=-1, x=6\) (b) No real solutions. Question 2: \[x=\frac{1}{3}, x=\frac{2}{3}\] Question 3: \[x=\frac{-5-\sqrt{109}}{6}, x=\frac{-5+\sqrt{109}}{6}\] Question 4: (a) \(x=3-\sqrt{7}, x=3+\sqrt{7}\) (b) \(x=\frac{-5-\sqrt{13}}{6}, x=\frac{-5+\sqrt{13}}{6}\) (c) \(x=4-\sqrt{19},x=4+\sqrt{19}\) |
The general quadratic formula - Worksheet A
Worksheet coming soon.
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PatrickJMT - The general quadratic formula:
PatrickJMT: How to derive the general quadratic formula
The general quadratic formula can be derived by completing the square.
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