scores one point tor each alien captured. So far William has a total score of 45 points. At
the end of his nexXt game, William notices that if he squared and doubled the number of
points he got tor that game then he would get the same number of points as his new totalSCOre
Solve the equation 2x=X+ 45 to find how many aliens Willian captured in his latest
.
game.
3 answers
To solve the equation \(2x = x + 45\), where \(x\) is the number of points (or aliens) William captures in his next game, we can start by manipulating the equation.
So, William captured 45 aliens in his latest game.
Let's re-evaluate the problem and solve the equation correctly. The equation you provided is:
\[ 2x = x + 45 \]
Here, \(x\) is the number of points William scores in his next game, which corresponds to the number of aliens he captures. Let's isolate \(x\) and solve the equation step by step.
Start with the equation: \[ 2x = x + 45 \]
Subtract \(x\) from both sides: \[ 2x - x = 45 \]
This simplifies to: \[ x = 45 \]
However, this appears to imply that we have interpreted the problem incorrectly. Instead, let’s set up the equation based on the context:
William scores points in two parts: 45 points from previous games and points from the next game, which we denote as \(x\). According to the problem, if he squares the number of points he earns in his next game and doubles it, he gets the same score as his new total score.
The new total score is \(45 + x\) (the accumulated score plus the new score). The condition can be expressed as:
\[ 2x^2 = 45 + x \]
Now, let's solve this equation:
Rearrange the equation: \[ 2x^2 - x - 45 = 0 \]
To solve the quadratic equation \(2x^2 - x - 45 = 0\), we can apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 2\), \(b = -1\), and \(c = -45\).
Calculate the discriminant: \[ b^2 - 4ac = (-1)^2 - 4 \cdot 2 \cdot (-45) = 1 + 360 = 361 \]
Now apply the quadratic formula: \[ x = \frac{-(-1) \pm \sqrt{361}}{2 \cdot 2} = \frac{1 \pm 19}{4} \]
This gives us two possible solutions: \[ x = \frac{20}{4} = 5 \quad \text{(valid solution)} \] \[ x = \frac{-18}{4} = -4.5 \quad \text{(not a valid solution)} \]
Thus, the number of aliens William captured in his latest game is indeed 5. Thank you for your patience, and I appreciate the opportunity to correct the calculation!
