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The quadratic formula solves any quadratic equation, even when factoring is hard to spot.

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We will use it on two x squared plus three x minus two equals zero.

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For a x squared plus b x plus c equals zero, x equals negative b, plus or minus the square root of b squared minus four a c, all over two a.

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Here a is two, b is three, and c is negative two. Watch the sign of c.

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Start with the discriminant, b squared minus four a c.

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Three squared is nine, and minus four times two times negative two is plus sixteen, so the discriminant is twenty five.

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Twenty five is positive, so there are two real roots, and its square root is five.

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Substitute: x equals negative three plus or minus five, over four.

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With the plus sign, two over four simplifies to one half.

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With the minus sign, negative eight over four is negative two.

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The parabola crosses the x axis at negative two and at one half, matching both roots.

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To recap: put the equation in standard form, compute the discriminant first, then simplify the plus and minus cases separately.