Question
\[\nx^{2}-10 x+21=0 \]
Ask by Macdonald Whittaker. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = 3 \) and \( x = 7 \).
Solution
To solve the quadratic equation:
\[
x^{2} - 10x + 21 = 0
\]
we can use factoring as follows.
**Step 1: Factor the quadratic**
We look for two numbers that multiply to \(21\) and add up to \(-10\). These numbers are \(-3\) and \(-7\), since:
\[
(-3) \times (-7) = 21 \quad \text{and} \quad (-3) + (-7) = -10
\]
So, the equation can be factored as:
\[
(x - 3)(x - 7) = 0
\]
**Step 2: Apply the Zero Product Property**
If the product of two factors is zero, at least one of the factors must be zero. Therefore, set each factor equal to zero:
\[
x - 3 = 0 \quad \text{or} \quad x - 7 = 0
\]
**Step 3: Solve for \(x\)**
\[
x = 3 \quad \text{or} \quad x = 7
\]
**Conclusion**
The solutions to the equation \(x^{2} - 10x + 21 = 0\) are:
\[
x = 3 \quad \text{and} \quad x = 7
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To solve the quadratic equation \(x^{2}-10x+21=0\), we can factor it. The equation can be rewritten as \((x-3)(x-7)=0\). Setting each factor to zero gives us the solutions \(x = 3\) and \(x = 7\). So, the roots of the equation are 3 and 7. If you ever find yourself stuck on a quadratic equation, remember the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). It works for any quadratic equation and is especially handy when factoring seems tricky! Just plug in your values for \(a\), \(b\), and \(c\) to calculate the roots. Common mistake? Forgetting to simplify the square root or getting the signs mixed up can lead to all sorts of mix-ups!
