Solve the equation 25x^2 + 10x = 6.

Step-by-step solution

1. 1

   Write the equation in standard form. Subtract $6$ from both sides:

   $25x^2+10x-6=0$

   so $a=25$, $b=10$, $c=-6$.

2. 2

   Compute the discriminant.

   $\Delta=b^2-4ac=100-4(25)(-6)=100+600=700$

   It is positive but not a perfect square ($26^2=676$, $27^2=729$), so expect two irrational roots and no integer factorisation.

3. 3

   Apply the quadratic formula.

   $x=\frac{-10\pm\sqrt{700}}{50}$

4. 4

   Simplify the radical, then the fraction. Pull the largest perfect square out of $700=100\cdot 7$:

   $\sqrt{700}=10\sqrt7\quad\Longrightarrow\quad x=\frac{-10\pm 10\sqrt7}{50}=\frac{-1\pm\sqrt7}{5}$

   Every term in the numerator and denominator shares the factor $10$, so cancelling is valid.

5. 5

   Check numerically. $\sqrt7\approx 2.6458$, giving $x\approx 0.3292$ or $x\approx-0.7292$. Then $25(0.3292)^2+10(0.3292)\approx 2.709+3.292\approx 6.00$ $\checkmark$.