Use the quadratic formula to solve the quadratic equation 2x2−9x+11=0

Question

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Express its solutions in the form a±bi
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Answer 1

To solve the quadratic equation $2x^2-9x+11=0$ using the quadratic formula, we start with the formula itself:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

In our equation, the coefficients are:

• $a=2$
• $b=-9$
• $c=11$

Let's first calculate the discriminant $b^2-4ac$:

$$b^2=(-9{)}^2=81$$ $$4ac=4\times 2\times 11=88$$ $$\text{Discriminant}=b^2-4ac=81-88=-7$$

Since the discriminant is negative ($-7$), this means the solutions will involve imaginary numbers.

Next, we substitute the values into the quadratic formula:

$$x=\frac{-(-9)\pm \sqrt{-7}}{2\cdot 2}$$ $$x=\frac{9\pm \sqrt{-7}}{4}$$

Now, let's express $\sqrt{-7}$ in terms of $i$:

$$\sqrt{-7}=i\sqrt{7}$$

Now we substitute this back into the formula:

$$x=\frac{9\pm i\sqrt{7}}{4}$$

We can separate this into real and imaginary parts:

$$x=\frac{9}{4}\pm \frac{i\sqrt{7}}{4}$$

Thus, the final solutions in the form $a\pm bi$ are:

$$x=\frac{9}{4}\pm \frac{\sqrt{7}}{4}i$$

So, the solutions to the equation $2x^2-9x+11=0$ are:

$$x=\frac{9}{4}+\frac{\sqrt{7}}{4}i\text{and}x=\frac{9}{4}-\frac{\sqrt{7}}{4}i$$