1. Problem 7: If four times a whole number is subtracted from three times the square of the number, the result 15 is obtained. Find the number. Step 1: Let the whole number be $x$. Step 2: Form the equation from the problem statement: $$3x^2 - 4x = 15$$ Step 3: Rearrange to standard quadratic form: $$3x^2 - 4x - 15 = 0$$ Step 4: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, here $a=3$, $b=-4$, $c=-15$. Step 5: Calculate the discriminant: $$\Delta = (-4)^2 - 4 \times 3 \times (-15) = 16 + 180 = 196$$ Step 6: Calculate roots: $$x = \frac{4 \pm \sqrt{196}}{2 \times 3} = \frac{4 \pm 14}{6}$$ Step 7: Possible values: - $x = \frac{4 + 14}{6} = \frac{18}{6} = 3$ - $x = \frac{4 - 14}{6} = \frac{-10}{6} = -\frac{5}{3}$ (not a whole number) Answer: The number is 3. 2. Problem 8: Two consecutive positive numbers are such that the sum of their squares is 113. Find the two numbers. Step 1: Let the numbers be $x$ and $x+1$ (consecutive integers). Step 2: Write equation for sum of squares: $$x^2 + (x+1)^2 = 113$$ Step 3: Expand: $$x^2 + x^2 + 2x + 1 = 113$$ Step 4: Simplify: $$2x^2 + 2x + 1 = 113$$ Step 5: Rearrange: $$2x^2 + 2x + 1 - 113 = 0$$ $$2x^2 + 2x - 112 = 0$$ Step 6: Divide whole equation by 2: $$x^2 + x - 56 = 0$$ Step 7: Factor quadratic: $$(x + 8)(x - 7) = 0$$ Step 8: Solutions: - $x = -8$ (not positive) - $x = 7$ (positive) Step 9: Numbers are 7 and 8. Answer: The two numbers are 7 and 8. 3. Problem 9: The difference between two positive numbers is 7 and the square of their sum is 289. Find the two numbers. Step 1: Let the two numbers be $x$ and $y$ with $x > y$. Step 2: From the problem: $$x - y = 7$$ $$ (x + y)^2 = 289 $$ Step 3: Solve for $x$ from the first: $x = y + 7$ Step 4: Substitute into the second: $$ (y + 7 + y)^2 = 289 $$ $$ (2y + 7)^2 = 289 $$ Step 5: Take square root: $$ 2y + 7 = \pm 17 $$ Step 6: Consider $2y + 7 = 17$ $$ 2y = 10 \, \Rightarrow \, y=5 $$ Step 7: Calculate $x$: $$ x = 5 + 7 = 12 $$ Step 8: Check $2y + 7 = -17$ $$ 2y = -24 \, \Rightarrow \, y = -12 $$ (not positive, discard) Answer: The two numbers are 12 and 5.