25x²+20x+4=(x-7)²
To solve the equation \( 25x^2 + 20x + 4 = (x - 7)^2 \), let's first expand the right side of the equation.
Expand \( (x - 7)^2 \): \[ (x - 7)^2 = x^2 - 14x + 49 \]
Now, substitute this back into the equation: \[ 25x^2 + 20x + 4 = x^2 - 14x + 49 \]
Next, move all terms to one side of the equation to set it to zero: \[ 25x^2 + 20x + 4 - x^2 + 14x - 49 = 0 \]
Combine like terms: \[ (25x^2 - x^2) + (20x + 14x) + (4 - 49) = 0 \] \[ 24x^2 + 34x - 45 = 0 \]
Now, we can simplify the equation: \[ 12x^2 + 17x - \frac{45}{2} = 0 \] (Or you can keep it as \( 24x^2 + 34x - 45 = 0 \) if you prefer not to use fractions.)
To solve the quadratic equation, we can apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 24\), \(b = 34\), and \(c = -45\).
Calculate the discriminant: \[ b^2 - 4ac = 34^2 - 4 \cdot 24 \cdot (-45) = 1156 + 4320 = 5476 \]
Now calculate the value of \(x\): \[ x = \frac{-34 \pm \sqrt{5476}}{2 \cdot 24} \] \[ \sqrt{5476} = 74 \] \[ x = \frac{-34 \pm 74}{48} \]
This gives us two possible solutions: \[ x_1 = \frac{-34 + 74}{48} = \frac{40}{48} = \frac{5}{6} \] \[ x_2 = \frac{-34 - 74}{48} = \frac{-108}{48} = -\frac{9}{4} \]
So the solutions to the equation \( 25x^2 + 20x + 4 = (x - 7)^2 \) are \( x = \frac{5}{6} \) and \( x = -\frac{9}{4} \).
