To solve the problem of finding two numbers that multiply to 225 and add to -34, let's break it down:

You are looking for two numbers, say $x$ and $y$, such that:

1. $x \times y = 225$
2. $x + y = -34$

Step-by-Step Approach:

1. Set up the equations: $x \times y = 225 \quad \text{(Multiplication condition)}$ $x + y = -34 \quad \text{(Addition condition)}$

2. Substitute one equation into the other: From the addition equation, solve for one variable in terms of the other. Let's solve for $y$: $y = -34 - x$ Now substitute this into the multiplication equation: $x \times (-34 - x) = 225$

3. Simplify and solve the quadratic equation: Distribute $x$ on the left-hand side: $-34x - x^2 = 225$ Rearrange the equation to get everything on one side: $x^2 + 34x - 225 = 0$ This is a quadratic equation.

4. Solve the quadratic equation: You can solve it using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $x^2 + 34x - 225 = 0$, the coefficients are:

   • $a = 1$
   • $b = 34$
   • $c = -225$

   Substitute these values into the quadratic formula: $x = \frac{-34 \pm \sqrt{34^2 - 4(1)(-225)}}{2(1)}$ $x = \frac{-34 \pm \sqrt{1156 + 900}}{2}$ $x = \frac{-34 \pm \sqrt{2056}}{2}$ $x = \frac{-34 \pm 45.33}{2}$

5. Find the two solutions: Now, calculate the two possible values of $x$: $x_1 = \frac{-34 + 45.33}{2} = \frac{11.33}{2} = 5.665$ $x_2 = \frac{-34 - 45.33}{2} = \frac{-79.33}{2} = -39.665$

6. Find the corresponding values of $y$: For $x_1 = 5.665$, $y = -34 - 5.665 = -39.665$. For $x_2 = -39.665$, $y = -34 - (-39.665) = 5.665$.

Conclusion:

The two numbers are approximately 5.665 and -39.665.

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Related Questions:

1. How do you apply the quadratic formula to solve real-world problems?
2. What are some shortcuts for factoring quadratic equations?
3. How do you verify the solution of a quadratic equation?
4. How can you simplify radicals in quadratic equations?
5. What are alternative methods to solve quadratic equations besides the quadratic formula?

Tip: When solving quadratic equations, always check if you can factor the equation before using the quadratic formula—it can save time!