1. **Problem:** Solve the quadratic equation $x^2 - 2x - 15 = 0$. 2. **Formula:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $a=1$, $b=-2$, and $c=-15$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4(1)(-15) = 4 + 60 = 64$$ 5. **Find the roots:** $$x = \frac{-(-2) \pm \sqrt{64}}{2(1)} = \frac{2 \pm 8}{2}$$ 6. **Evaluate each root:** - For $+$ sign: $x = \frac{2 + 8}{2} = \frac{10}{2} = 5$ - For $-$ sign: $x = \frac{2 - 8}{2} = \frac{-6}{2} = -3$ 7. **Answer:** The solutions to the quadratic equation are $x = 5$ and $x = -3$. --- 1. **Problem:** Make $t$ the subject of the formula $v = u + at$. 2. **Rearrange the formula:** $$v = u + at \implies at = v - u$$ 3. **Solve for $t$:** $$t = \frac{v - u}{a}$$ 4. **Answer:** $t = \frac{v - u}{a}$