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How to Solve the Quadratic Equation $x^2 + 12x - 8 = 0$
Mathematics
Grade 10 (Junior High School)
Question Content
Solve the quadratic equation: $x^2 + 12x - 8 = 0$
Correct Answer
$x = -6 + 2\\sqrt{11}$ or $x = -6 - 2\\sqrt{11}$
Detailed Solution Steps
1
Step 1: Identify the coefficients in the standard quadratic form $ax^2+bx+c=0$. Here, $a=1$, $b=12$, $c=-8$.
2
Step 2: Use the quadratic formula $x = \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$, which is used to solve any quadratic equation when factoring is not straightforward.
3
Step 3: Calculate the discriminant first: $\\Delta = b^2-4ac = 12^2 - 4\\times1\\times(-8) = 144 + 32 = 176$.
4
Step 4: Substitute the values into the quadratic formula: $x = \\frac{-12\\pm\\sqrt{176}}{2\\times1}$. Simplify $\\sqrt{176} = \\sqrt{16\\times11} = 4\\sqrt{11}$.
5
Step 5: Simplify the expression: $x = \\frac{-12\\pm4\\sqrt{11}}{2} = -6\\pm2\\sqrt{11}$. This gives the two solutions: $x = -6 + 2\\sqrt{11}$ and $x = -6 - 2\\sqrt{11}$.
Knowledge Points Involved
1
Standard form of a quadratic equation
A quadratic equation is written as $ax^2+bx+c=0$ where $a$, $b$, $c$ are constants and $a\\neq0$. This form is required to use methods like the quadratic formula or completing the square, as it clearly defines the coefficients needed for these techniques.
2
Quadratic formula
The formula $x = \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ is derived from completing the square for the standard quadratic equation. It can solve any quadratic equation, regardless of whether it can be factored. The $\\pm$ sign accounts for the two possible solutions of a quadratic equation.
3
Discriminant of a quadratic equation
The discriminant is $\\Delta = b^2-4ac$. It determines the nature of the roots: if $\\Delta>0$, there are two distinct real roots; if $\\Delta=0$, there is one real repeated root; if $\\Delta<0$, there are two complex conjugate roots. For this problem, $\\Delta=176>0$, so there are two distinct real solutions.
4
Simplification of radical expressions
To simplify square roots, factor the radicand into a product of a perfect square and another integer. For $\\sqrt{176}$, we factor it as $\\sqrt{16\\times11} = \\sqrt{16}\\times\\sqrt{11}=4\\sqrt{11}$, which makes the final solution in its simplest radical form.
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