What is the solution to the equation x^2 + 6x = 16?
Understand the Problem
The question is asking for the solution of a quadratic equation. We can solve it by rearranging the equation into standard form and using the quadratic formula or factoring to find the value of x.
Answer
The solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Answer for screen readers
The solutions to the quadratic equation, based on the values of $a$, $b$, and $c$, will be:
$$ x_1, x_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Steps to Solve
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Rearranging the Equation First, ensure the quadratic equation is in standard form, which is $ax^2 + bx + c = 0$. If the equation is not in this form, rearrange it appropriately.
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Identifying Coefficients Identify the coefficients $a$, $b$, and $c$ from the standard form of the equation. These are necessary for applying the quadratic formula.
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Applying the Quadratic Formula Use the quadratic formula to find the values of $x$: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ This formula will provide the solutions for the quadratic equation.
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Calculating the Discriminant Calculate the discriminant $D = b^2 - 4ac$. This value helps determine the nature of the roots:
- If $D > 0$, there are two distinct real solutions.
- If $D = 0$, there is one real solution (a repeated root).
- If $D < 0$, there are no real solutions (complex roots).
- Finding the Solutions Substitute the values of $a$, $b$, and $c$ into the quadratic formula. Calculate both possible values for $x$ depending on $D$.
The solutions to the quadratic equation, based on the values of $a$, $b$, and $c$, will be:
$$ x_1, x_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
More Information
The quadratic formula is a powerful tool that can solve any quadratic equation, regardless of whether it can be factored easily. Quadratic equations are common in both pure and applied mathematics, and understanding their solutions is key to mastering algebra.
Tips
- Forgetting to set the equation to zero before identifying coefficients.
- Miscalculating the discriminant, which can lead to incorrect conclusions about the number of solutions.
- Not simplifying the square root correctly when solving for $x$.
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