The sum of a number and its square is 42.
Understand the Problem
The question is asking us to find a number such that when we add the number to its square, the result is 42. This implies solving the equation x + x² = 42, where x is the unknown number.
Answer
$x = 6$ and $x = -7$
Answer for screen readers
The solutions to the problem are $x = 6$ and $x = -7$.
Steps to Solve
- Set up the equation
We start with the equation derived from the problem statement: $$ x + x^2 = 42 $$
- Rearrange the equation
We rearrange this equation to set it to zero: $$ x^2 + x - 42 = 0 $$
- Identify coefficients
In the quadratic equation $ax^2 + bx + c = 0$, here:
- $a = 1$
- $b = 1$
- $c = -42$
- Use the quadratic formula
We can now apply the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Substituting in our values: $$ x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)} $$
- Calculate the discriminant
Calculate the value inside the square root (the discriminant): $$ 1^2 - 4(1)(-42) = 1 + 168 = 169 $$
- Find the roots
Now substitute the discriminant back into the quadratic formula: $$ x = \frac{-1 \pm \sqrt{169}}{2} $$
Since $ \sqrt{169} = 13$, we have: $$ x = \frac{-1 \pm 13}{2} $$
This gives us two possible solutions:
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$$ x = \frac{12}{2} = 6 $$
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$$ x = \frac{-14}{2} = -7 $$
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List the final answers
The two possible numbers satisfying the condition are: $$ x = 6 \text{ and } x = -7 $$
The solutions to the problem are $x = 6$ and $x = -7$.
More Information
Both $6$ and $-7$ satisfy the original equation $x + x^2 = 42$:
- For $6$: $6 + 6^2 = 6 + 36 = 42$.
- For $-7$: $-7 + (-7)^2 = -7 + 49 = 42$.
Tips
- Forgetting to set the equation to zero before applying the quadratic formula.
- Miscalculating the discriminant, leading to incorrect roots.
